Lagrange four-square (Bachet Conjecture) for 2023

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2023

Lagrange Four Square Definition

For any natural number (p), we write as

p = a² + b² + c² + d²

Determine max_a:

Floor(√2023) = Floor(44.977772288098)

Floor(44.977772288098) = 44
This is called max_a

Determine min_a:

Find the first value of a such that
a² ≥ n/4

Start with min_a = 0 and increase by 1

Continue until we reach or breach n/4 → 2023/4 = 505.75

When min_a = 23, then it is a² = 529 ≥ 505.75, so min_a = 23

Find a in the range of (min_a, max_a)

(0, 44)

a = 0

Find max_b which is Floor(√n - a²)

max_b = Floor(√2023 - 0²)

max_b = Floor(√2023 - 0)

max_b = Floor(√2023)

max_b = Floor(44.977772288098)

max_b = 44

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 0 and increase by 1
Go until (n - a²)/3 → (2023 - 0²)/3 = 674.33333333333

When min_b = 26, then it is b² = 676 ≥ 674.33333333333, so min_b = 26

Test values for b in the range of (min_b, max_b)

(26, 44)

b = 26

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 26²)

max_c = Floor(√2023 - 0 - 676)

max_c = Floor(√1347)

max_c = Floor(36.701498607005)

max_c = 36

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 26²)/2 = 673.5

When min_c = 26, then it is c² = 676 ≥ 673.5, so min_c = 26

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 26² - 26²

max_d = √2023 - 0 - 676 - 676

max_d = √671

max_d = 25.903667693977

Since max_d = 25.903667693977 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 26² - 27²

max_d = √2023 - 0 - 676 - 729

max_d = √618

max_d = 24.859605789312

Since max_d = 24.859605789312 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 26² - 28²

max_d = √2023 - 0 - 676 - 784

max_d = √563

max_d = 23.727621035409

Since max_d = 23.727621035409 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 26² - 29²

max_d = √2023 - 0 - 676 - 841

max_d = √506

max_d = 22.494443758404

Since max_d = 22.494443758404 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 26² - 30²

max_d = √2023 - 0 - 676 - 900

max_d = √447

max_d = 21.142374511866

Since max_d = 21.142374511866 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 26² - 31²

max_d = √2023 - 0 - 676 - 961

max_d = √386

max_d = 19.646882704388

Since max_d = 19.646882704388 is not an integer, this is not a solution

c = 32

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 26² - 32²

max_d = √2023 - 0 - 676 - 1024

max_d = √323

max_d = 17.972200755611

Since max_d = 17.972200755611 is not an integer, this is not a solution

c = 33

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 26² - 33²

max_d = √2023 - 0 - 676 - 1089

max_d = √258

max_d = 16.062378404209

Since max_d = 16.062378404209 is not an integer, this is not a solution

c = 34

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 26² - 34²

max_d = √2023 - 0 - 676 - 1156

max_d = √191

max_d = 13.820274961085

Since max_d = 13.820274961085 is not an integer, this is not a solution

c = 35

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 26² - 35²

max_d = √2023 - 0 - 676 - 1225

max_d = √122

max_d = 11.045361017187

Since max_d = 11.045361017187 is not an integer, this is not a solution

c = 36

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 26² - 36²

max_d = √2023 - 0 - 676 - 1296

max_d = √51

max_d = 7.1414284285429

Since max_d = 7.1414284285429 is not an integer, this is not a solution

b = 27

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 27²)

max_c = Floor(√2023 - 0 - 729)

max_c = Floor(√1294)

max_c = Floor(35.97221149721)

max_c = 35

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 27²)/2 = 647

When min_c = 26, then it is c² = 676 ≥ 647, so min_c = 26

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 27² - 26²

max_d = √2023 - 0 - 729 - 676

max_d = √618

max_d = 24.859605789312

Since max_d = 24.859605789312 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 27² - 27²

max_d = √2023 - 0 - 729 - 729

max_d = √565

max_d = 23.769728648009

Since max_d = 23.769728648009 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 27² - 28²

max_d = √2023 - 0 - 729 - 784

max_d = √510

max_d = 22.583179581272

Since max_d = 22.583179581272 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 27² - 29²

max_d = √2023 - 0 - 729 - 841

max_d = √453

max_d = 21.283796653793

Since max_d = 21.283796653793 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 27² - 30²

max_d = √2023 - 0 - 729 - 900

max_d = √394

max_d = 19.849433241279

Since max_d = 19.849433241279 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 27² - 31²

max_d = √2023 - 0 - 729 - 961

max_d = √333

max_d = 18.248287590895

Since max_d = 18.248287590895 is not an integer, this is not a solution

c = 32

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 27² - 32²

max_d = √2023 - 0 - 729 - 1024

max_d = √270

max_d = 16.431676725155

Since max_d = 16.431676725155 is not an integer, this is not a solution

c = 33

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 27² - 33²

max_d = √2023 - 0 - 729 - 1089

max_d = √205

max_d = 14.317821063276

Since max_d = 14.317821063276 is not an integer, this is not a solution

c = 34

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 27² - 34²

max_d = √2023 - 0 - 729 - 1156

max_d = √138

max_d = 11.747340124471

Since max_d = 11.747340124471 is not an integer, this is not a solution

c = 35

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 27² - 35²

max_d = √2023 - 0 - 729 - 1225

max_d = √69

max_d = 8.3066238629181

Since max_d = 8.3066238629181 is not an integer, this is not a solution

b = 28

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 28²)

max_c = Floor(√2023 - 0 - 784)

max_c = Floor(√1239)

max_c = Floor(35.199431813596)

max_c = 35

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 28²)/2 = 619.5

When min_c = 25, then it is c² = 625 ≥ 619.5, so min_c = 25

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 28² - 25²

max_d = √2023 - 0 - 784 - 625

max_d = √614

max_d = 24.779023386728

Since max_d = 24.779023386728 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 28² - 26²

max_d = √2023 - 0 - 784 - 676

max_d = √563

max_d = 23.727621035409

Since max_d = 23.727621035409 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 28² - 27²

max_d = √2023 - 0 - 784 - 729

max_d = √510

max_d = 22.583179581272

Since max_d = 22.583179581272 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 28² - 28²

max_d = √2023 - 0 - 784 - 784

max_d = √455

max_d = 21.330729007702

Since max_d = 21.330729007702 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 28² - 29²

max_d = √2023 - 0 - 784 - 841

max_d = √398

max_d = 19.94993734326

Since max_d = 19.94993734326 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 28² - 30²

max_d = √2023 - 0 - 784 - 900

max_d = √339

max_d = 18.411952639522

Since max_d = 18.411952639522 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 28² - 31²

max_d = √2023 - 0 - 784 - 961

max_d = √278

max_d = 16.673332000533

Since max_d = 16.673332000533 is not an integer, this is not a solution

c = 32

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 28² - 32²

max_d = √2023 - 0 - 784 - 1024

max_d = √215

max_d = 14.662878298615

Since max_d = 14.662878298615 is not an integer, this is not a solution

c = 33

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 28² - 33²

max_d = √2023 - 0 - 784 - 1089

max_d = √150

max_d = 12.247448713916

Since max_d = 12.247448713916 is not an integer, this is not a solution

c = 34

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 28² - 34²

max_d = √2023 - 0 - 784 - 1156

max_d = √83

max_d = 9.1104335791443

Since max_d = 9.1104335791443 is not an integer, this is not a solution

c = 35

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 28² - 35²

max_d = √2023 - 0 - 784 - 1225

max_d = √14

max_d = 3.7416573867739

Since max_d = 3.7416573867739 is not an integer, this is not a solution

b = 29

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 29²)

max_c = Floor(√2023 - 0 - 841)

max_c = Floor(√1182)

max_c = Floor(34.380226875342)

max_c = 34

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 29²)/2 = 591

When min_c = 25, then it is c² = 625 ≥ 591, so min_c = 25

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 29² - 25²

max_d = √2023 - 0 - 841 - 625

max_d = √557

max_d = 23.600847442412

Since max_d = 23.600847442412 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 29² - 26²

max_d = √2023 - 0 - 841 - 676

max_d = √506

max_d = 22.494443758404

Since max_d = 22.494443758404 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 29² - 27²

max_d = √2023 - 0 - 841 - 729

max_d = √453

max_d = 21.283796653793

Since max_d = 21.283796653793 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 29² - 28²

max_d = √2023 - 0 - 841 - 784

max_d = √398

max_d = 19.94993734326

Since max_d = 19.94993734326 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 29² - 29²

max_d = √2023 - 0 - 841 - 841

max_d = √341

max_d = 18.466185312619

Since max_d = 18.466185312619 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 29² - 30²

max_d = √2023 - 0 - 841 - 900

max_d = √282

max_d = 16.792855623747

Since max_d = 16.792855623747 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 29² - 31²

max_d = √2023 - 0 - 841 - 961

max_d = √221

max_d = 14.866068747319

Since max_d = 14.866068747319 is not an integer, this is not a solution

c = 32

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 29² - 32²

max_d = √2023 - 0 - 841 - 1024

max_d = √158

max_d = 12.569805089977

Since max_d = 12.569805089977 is not an integer, this is not a solution

c = 33

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 29² - 33²

max_d = √2023 - 0 - 841 - 1089

max_d = √93

max_d = 9.643650760993

Since max_d = 9.643650760993 is not an integer, this is not a solution

c = 34

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 29² - 34²

max_d = √2023 - 0 - 841 - 1156

max_d = √26

max_d = 5.0990195135928

Since max_d = 5.0990195135928 is not an integer, this is not a solution

b = 30

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 30²)

max_c = Floor(√2023 - 0 - 900)

max_c = Floor(√1123)

max_c = Floor(33.511192160232)

max_c = 33

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 30²)/2 = 561.5

When min_c = 24, then it is c² = 576 ≥ 561.5, so min_c = 24

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 30² - 24²

max_d = √2023 - 0 - 900 - 576

max_d = √547

max_d = 23.388031127053

Since max_d = 23.388031127053 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 30² - 25²

max_d = √2023 - 0 - 900 - 625

max_d = √498

max_d = 22.315913604421

Since max_d = 22.315913604421 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 30² - 26²

max_d = √2023 - 0 - 900 - 676

max_d = √447

max_d = 21.142374511866

Since max_d = 21.142374511866 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 30² - 27²

max_d = √2023 - 0 - 900 - 729

max_d = √394

max_d = 19.849433241279

Since max_d = 19.849433241279 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 30² - 28²

max_d = √2023 - 0 - 900 - 784

max_d = √339

max_d = 18.411952639522

Since max_d = 18.411952639522 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 30² - 29²

max_d = √2023 - 0 - 900 - 841

max_d = √282

max_d = 16.792855623747

Since max_d = 16.792855623747 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 30² - 30²

max_d = √2023 - 0 - 900 - 900

max_d = √223

max_d = 14.933184523068

Since max_d = 14.933184523068 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 30² - 31²

max_d = √2023 - 0 - 900 - 961

max_d = √162

max_d = 12.727922061358

Since max_d = 12.727922061358 is not an integer, this is not a solution

c = 32

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 30² - 32²

max_d = √2023 - 0 - 900 - 1024

max_d = √99

max_d = 9.9498743710662

Since max_d = 9.9498743710662 is not an integer, this is not a solution

c = 33

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 30² - 33²

max_d = √2023 - 0 - 900 - 1089

max_d = √34

max_d = 5.8309518948453

Since max_d = 5.8309518948453 is not an integer, this is not a solution

b = 31

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 31²)

max_c = Floor(√2023 - 0 - 961)

max_c = Floor(√1062)

max_c = Floor(32.588341473601)

max_c = 32

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 31²)/2 = 531

When min_c = 24, then it is c² = 576 ≥ 531, so min_c = 24

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 31² - 24²

max_d = √2023 - 0 - 961 - 576

max_d = √486

max_d = 22.045407685049

Since max_d = 22.045407685049 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 31² - 25²

max_d = √2023 - 0 - 961 - 625

max_d = √437

max_d = 20.904544960367

Since max_d = 20.904544960367 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 31² - 26²

max_d = √2023 - 0 - 961 - 676

max_d = √386

max_d = 19.646882704388

Since max_d = 19.646882704388 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 31² - 27²

max_d = √2023 - 0 - 961 - 729

max_d = √333

max_d = 18.248287590895

Since max_d = 18.248287590895 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 31² - 28²

max_d = √2023 - 0 - 961 - 784

max_d = √278

max_d = 16.673332000533

Since max_d = 16.673332000533 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 31² - 29²

max_d = √2023 - 0 - 961 - 841

max_d = √221

max_d = 14.866068747319

Since max_d = 14.866068747319 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 31² - 30²

max_d = √2023 - 0 - 961 - 900

max_d = √162

max_d = 12.727922061358

Since max_d = 12.727922061358 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 31² - 31²

max_d = √2023 - 0 - 961 - 961

max_d = √101

max_d = 10.049875621121

Since max_d = 10.049875621121 is not an integer, this is not a solution

c = 32

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 31² - 32²

max_d = √2023 - 0 - 961 - 1024

max_d = √38

max_d = 6.164414002969

Since max_d = 6.164414002969 is not an integer, this is not a solution

b = 32

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 32²)

max_c = Floor(√2023 - 0 - 1024)

max_c = Floor(√999)

max_c = Floor(31.606961258558)

max_c = 31

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 32²)/2 = 499.5

When min_c = 23, then it is c² = 529 ≥ 499.5, so min_c = 23

c = 23

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 32² - 23²

max_d = √2023 - 0 - 1024 - 529

max_d = √470

max_d = 21.679483388679

Since max_d = 21.679483388679 is not an integer, this is not a solution

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 32² - 24²

max_d = √2023 - 0 - 1024 - 576

max_d = √423

max_d = 20.566963801203

Since max_d = 20.566963801203 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 32² - 25²

max_d = √2023 - 0 - 1024 - 625

max_d = √374

max_d = 19.339079605814

Since max_d = 19.339079605814 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 32² - 26²

max_d = √2023 - 0 - 1024 - 676

max_d = √323

max_d = 17.972200755611

Since max_d = 17.972200755611 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 32² - 27²

max_d = √2023 - 0 - 1024 - 729

max_d = √270

max_d = 16.431676725155

Since max_d = 16.431676725155 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 32² - 28²

max_d = √2023 - 0 - 1024 - 784

max_d = √215

max_d = 14.662878298615

Since max_d = 14.662878298615 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 32² - 29²

max_d = √2023 - 0 - 1024 - 841

max_d = √158

max_d = 12.569805089977

Since max_d = 12.569805089977 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 32² - 30²

max_d = √2023 - 0 - 1024 - 900

max_d = √99

max_d = 9.9498743710662

Since max_d = 9.9498743710662 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 32² - 31²

max_d = √2023 - 0 - 1024 - 961

max_d = √38

max_d = 6.164414002969

Since max_d = 6.164414002969 is not an integer, this is not a solution

b = 33

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 33²)

max_c = Floor(√2023 - 0 - 1089)

max_c = Floor(√934)

max_c = Floor(30.561413579872)

max_c = 30

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 33²)/2 = 467

When min_c = 22, then it is c² = 484 ≥ 467, so min_c = 22

c = 22

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 33² - 22²

max_d = √2023 - 0 - 1089 - 484

max_d = √450

max_d = 21.213203435596

Since max_d = 21.213203435596 is not an integer, this is not a solution

c = 23

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 33² - 23²

max_d = √2023 - 0 - 1089 - 529

max_d = √405

max_d = 20.124611797498

Since max_d = 20.124611797498 is not an integer, this is not a solution

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 33² - 24²

max_d = √2023 - 0 - 1089 - 576

max_d = √358

max_d = 18.920887928425

Since max_d = 18.920887928425 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 33² - 25²

max_d = √2023 - 0 - 1089 - 625

max_d = √309

max_d = 17.578395831247

Since max_d = 17.578395831247 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 33² - 26²

max_d = √2023 - 0 - 1089 - 676

max_d = √258

max_d = 16.062378404209

Since max_d = 16.062378404209 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 33² - 27²

max_d = √2023 - 0 - 1089 - 729

max_d = √205

max_d = 14.317821063276

Since max_d = 14.317821063276 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 33² - 28²

max_d = √2023 - 0 - 1089 - 784

max_d = √150

max_d = 12.247448713916

Since max_d = 12.247448713916 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 33² - 29²

max_d = √2023 - 0 - 1089 - 841

max_d = √93

max_d = 9.643650760993

Since max_d = 9.643650760993 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 33² - 30²

max_d = √2023 - 0 - 1089 - 900

max_d = √34

max_d = 5.8309518948453

Since max_d = 5.8309518948453 is not an integer, this is not a solution

b = 34

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 34²)

max_c = Floor(√2023 - 0 - 1156)

max_c = Floor(√867)

max_c = Floor(29.444863728671)

max_c = 29

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 34²)/2 = 433.5

When min_c = 21, then it is c² = 441 ≥ 433.5, so min_c = 21

c = 21

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 34² - 21²

max_d = √2023 - 0 - 1156 - 441

max_d = √426

max_d = 20.63976744055

Since max_d = 20.63976744055 is not an integer, this is not a solution

c = 22

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 34² - 22²

max_d = √2023 - 0 - 1156 - 484

max_d = √383

max_d = 19.570385790781

Since max_d = 19.570385790781 is not an integer, this is not a solution

c = 23

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 34² - 23²

max_d = √2023 - 0 - 1156 - 529

max_d = √338

max_d = 18.38477631085

Since max_d = 18.38477631085 is not an integer, this is not a solution

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 34² - 24²

max_d = √2023 - 0 - 1156 - 576

max_d = √291

max_d = 17.058722109232

Since max_d = 17.058722109232 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 34² - 25²

max_d = √2023 - 0 - 1156 - 625

max_d = √242

max_d = 15.556349186104

Since max_d = 15.556349186104 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 34² - 26²

max_d = √2023 - 0 - 1156 - 676

max_d = √191

max_d = 13.820274961085

Since max_d = 13.820274961085 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 34² - 27²

max_d = √2023 - 0 - 1156 - 729

max_d = √138

max_d = 11.747340124471

Since max_d = 11.747340124471 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 34² - 28²

max_d = √2023 - 0 - 1156 - 784

max_d = √83

max_d = 9.1104335791443

Since max_d = 9.1104335791443 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 34² - 29²

max_d = √2023 - 0 - 1156 - 841

max_d = √26

max_d = 5.0990195135928

Since max_d = 5.0990195135928 is not an integer, this is not a solution

b = 35

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 35²)

max_c = Floor(√2023 - 0 - 1225)

max_c = Floor(√798)

max_c = Floor(28.248893783651)

max_c = 28

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 35²)/2 = 399

When min_c = 20, then it is c² = 400 ≥ 399, so min_c = 20

c = 20

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 35² - 20²

max_d = √2023 - 0 - 1225 - 400

max_d = √398

max_d = 19.94993734326

Since max_d = 19.94993734326 is not an integer, this is not a solution

c = 21

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 35² - 21²

max_d = √2023 - 0 - 1225 - 441

max_d = √357

max_d = 18.894443627691

Since max_d = 18.894443627691 is not an integer, this is not a solution

c = 22

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 35² - 22²

max_d = √2023 - 0 - 1225 - 484

max_d = √314

max_d = 17.720045146669

Since max_d = 17.720045146669 is not an integer, this is not a solution

c = 23

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 35² - 23²

max_d = √2023 - 0 - 1225 - 529

max_d = √269

max_d = 16.401219466857

Since max_d = 16.401219466857 is not an integer, this is not a solution

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 35² - 24²

max_d = √2023 - 0 - 1225 - 576

max_d = √222

max_d = 14.899664425751

Since max_d = 14.899664425751 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 35² - 25²

max_d = √2023 - 0 - 1225 - 625

max_d = √173

max_d = 13.152946437966

Since max_d = 13.152946437966 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 35² - 26²

max_d = √2023 - 0 - 1225 - 676

max_d = √122

max_d = 11.045361017187

Since max_d = 11.045361017187 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 35² - 27²

max_d = √2023 - 0 - 1225 - 729

max_d = √69

max_d = 8.3066238629181

Since max_d = 8.3066238629181 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 35² - 28²

max_d = √2023 - 0 - 1225 - 784

max_d = √14

max_d = 3.7416573867739

Since max_d = 3.7416573867739 is not an integer, this is not a solution

b = 36

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 36²)

max_c = Floor(√2023 - 0 - 1296)

max_c = Floor(√727)

max_c = Floor(26.962937525426)

max_c = 26

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 36²)/2 = 363.5

When min_c = 20, then it is c² = 400 ≥ 363.5, so min_c = 20

c = 20

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 36² - 20²

max_d = √2023 - 0 - 1296 - 400

max_d = √327

max_d = 18.083141320025

Since max_d = 18.083141320025 is not an integer, this is not a solution

c = 21

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 36² - 21²

max_d = √2023 - 0 - 1296 - 441

max_d = √286

max_d = 16.911534525288

Since max_d = 16.911534525288 is not an integer, this is not a solution

c = 22

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 36² - 22²

max_d = √2023 - 0 - 1296 - 484

max_d = √243

max_d = 15.58845726812

Since max_d = 15.58845726812 is not an integer, this is not a solution

c = 23

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 36² - 23²

max_d = √2023 - 0 - 1296 - 529

max_d = √198

max_d = 14.07124727947

Since max_d = 14.07124727947 is not an integer, this is not a solution

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 36² - 24²

max_d = √2023 - 0 - 1296 - 576

max_d = √151

max_d = 12.288205727445

Since max_d = 12.288205727445 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 36² - 25²

max_d = √2023 - 0 - 1296 - 625

max_d = √102

max_d = 10.099504938362

Since max_d = 10.099504938362 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 36² - 26²

max_d = √2023 - 0 - 1296 - 676

max_d = √51

max_d = 7.1414284285429

Since max_d = 7.1414284285429 is not an integer, this is not a solution

b = 37

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 37²)

max_c = Floor(√2023 - 0 - 1369)

max_c = Floor(√654)

max_c = Floor(25.573423705089)

max_c = 25

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 37²)/2 = 327

When min_c = 19, then it is c² = 361 ≥ 327, so min_c = 19

c = 19

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 37² - 19²

max_d = √2023 - 0 - 1369 - 361

max_d = √293

max_d = 17.117242768624

Since max_d = 17.117242768624 is not an integer, this is not a solution

c = 20

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 37² - 20²

max_d = √2023 - 0 - 1369 - 400

max_d = √254

max_d = 15.937377450509

Since max_d = 15.937377450509 is not an integer, this is not a solution

c = 21

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 37² - 21²

max_d = √2023 - 0 - 1369 - 441

max_d = √213

max_d = 14.594519519326

Since max_d = 14.594519519326 is not an integer, this is not a solution

c = 22

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 37² - 22²

max_d = √2023 - 0 - 1369 - 484

max_d = √170

max_d = 13.038404810405

Since max_d = 13.038404810405 is not an integer, this is not a solution

c = 23

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 37² - 23²

max_d = √2023 - 0 - 1369 - 529

max_d = √125

max_d = 11.180339887499

Since max_d = 11.180339887499 is not an integer, this is not a solution

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 37² - 24²

max_d = √2023 - 0 - 1369 - 576

max_d = √78

max_d = 8.8317608663278

Since max_d = 8.8317608663278 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 37² - 25²

max_d = √2023 - 0 - 1369 - 625

max_d = √29

max_d = 5.3851648071345

Since max_d = 5.3851648071345 is not an integer, this is not a solution

b = 38

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 38²)

max_c = Floor(√2023 - 0 - 1444)

max_c = Floor(√579)

max_c = Floor(24.062418831032)

max_c = 24

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 38²)/2 = 289.5

When min_c = 18, then it is c² = 324 ≥ 289.5, so min_c = 18

c = 18

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 38² - 18²

max_d = √2023 - 0 - 1444 - 324

max_d = √255

max_d = 15.968719422671

Since max_d = 15.968719422671 is not an integer, this is not a solution

c = 19

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 38² - 19²

max_d = √2023 - 0 - 1444 - 361

max_d = √218

max_d = 14.764823060233

Since max_d = 14.764823060233 is not an integer, this is not a solution

c = 20

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 38² - 20²

max_d = √2023 - 0 - 1444 - 400

max_d = √179

max_d = 13.37908816026

Since max_d = 13.37908816026 is not an integer, this is not a solution

c = 21

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 38² - 21²

max_d = √2023 - 0 - 1444 - 441

max_d = √138

max_d = 11.747340124471

Since max_d = 11.747340124471 is not an integer, this is not a solution

c = 22

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 38² - 22²

max_d = √2023 - 0 - 1444 - 484

max_d = √95

max_d = 9.746794344809

Since max_d = 9.746794344809 is not an integer, this is not a solution

c = 23

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 38² - 23²

max_d = √2023 - 0 - 1444 - 529

max_d = √50

max_d = 7.0710678118655

Since max_d = 7.0710678118655 is not an integer, this is not a solution

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 38² - 24²

max_d = √2023 - 0 - 1444 - 576

max_d = √3

max_d = 1.7320508075689

Since max_d = 1.7320508075689 is not an integer, this is not a solution

b = 39

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 39²)

max_c = Floor(√2023 - 0 - 1521)

max_c = Floor(√502)

max_c = Floor(22.405356502408)

max_c = 22

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 39²)/2 = 251

When min_c = 16, then it is c² = 256 ≥ 251, so min_c = 16

c = 16

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 39² - 16²

max_d = √2023 - 0 - 1521 - 256

max_d = √246

max_d = 15.684387141358

Since max_d = 15.684387141358 is not an integer, this is not a solution

c = 17

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 39² - 17²

max_d = √2023 - 0 - 1521 - 289

max_d = √213

max_d = 14.594519519326

Since max_d = 14.594519519326 is not an integer, this is not a solution

c = 18

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 39² - 18²

max_d = √2023 - 0 - 1521 - 324

max_d = √178

max_d = 13.341664064126

Since max_d = 13.341664064126 is not an integer, this is not a solution

c = 19

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 39² - 19²

max_d = √2023 - 0 - 1521 - 361

max_d = √141

max_d = 11.874342087038

Since max_d = 11.874342087038 is not an integer, this is not a solution

c = 20

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 39² - 20²

max_d = √2023 - 0 - 1521 - 400

max_d = √102

max_d = 10.099504938362

Since max_d = 10.099504938362 is not an integer, this is not a solution

c = 21

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 39² - 21²

max_d = √2023 - 0 - 1521 - 441

max_d = √61

max_d = 7.8102496759067

Since max_d = 7.8102496759067 is not an integer, this is not a solution

c = 22

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 39² - 22²

max_d = √2023 - 0 - 1521 - 484

max_d = √18

max_d = 4.2426406871193

Since max_d = 4.2426406871193 is not an integer, this is not a solution

b = 40

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 40²)

max_c = Floor(√2023 - 0 - 1600)

max_c = Floor(√423)

max_c = Floor(20.566963801203)

max_c = 20

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 40²)/2 = 211.5

When min_c = 15, then it is c² = 225 ≥ 211.5, so min_c = 15

c = 15

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 40² - 15²

max_d = √2023 - 0 - 1600 - 225

max_d = √198

max_d = 14.07124727947

Since max_d = 14.07124727947 is not an integer, this is not a solution

c = 16

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 40² - 16²

max_d = √2023 - 0 - 1600 - 256

max_d = √167

max_d = 12.92284798332

Since max_d = 12.92284798332 is not an integer, this is not a solution

c = 17

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 40² - 17²

max_d = √2023 - 0 - 1600 - 289

max_d = √134

max_d = 11.57583690279

Since max_d = 11.57583690279 is not an integer, this is not a solution

c = 18

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 40² - 18²

max_d = √2023 - 0 - 1600 - 324

max_d = √99

max_d = 9.9498743710662

Since max_d = 9.9498743710662 is not an integer, this is not a solution

c = 19

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 40² - 19²

max_d = √2023 - 0 - 1600 - 361

max_d = √62

max_d = 7.8740078740118

Since max_d = 7.8740078740118 is not an integer, this is not a solution

c = 20

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 40² - 20²

max_d = √2023 - 0 - 1600 - 400

max_d = √23

max_d = 4.7958315233127

Since max_d = 4.7958315233127 is not an integer, this is not a solution

b = 41

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 41²)

max_c = Floor(√2023 - 0 - 1681)

max_c = Floor(√342)

max_c = Floor(18.493242008907)

max_c = 18

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 41²)/2 = 171

When min_c = 14, then it is c² = 196 ≥ 171, so min_c = 14

c = 14

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 41² - 14²

max_d = √2023 - 0 - 1681 - 196

max_d = √146

max_d = 12.083045973595

Since max_d = 12.083045973595 is not an integer, this is not a solution

c = 15

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 41² - 15²

max_d = √2023 - 0 - 1681 - 225

max_d = √117

max_d = 10.816653826392

Since max_d = 10.816653826392 is not an integer, this is not a solution

c = 16

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 41² - 16²

max_d = √2023 - 0 - 1681 - 256

max_d = √86

max_d = 9.2736184954957

Since max_d = 9.2736184954957 is not an integer, this is not a solution

c = 17

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 41² - 17²

max_d = √2023 - 0 - 1681 - 289

max_d = √53

max_d = 7.2801098892805

Since max_d = 7.2801098892805 is not an integer, this is not a solution

c = 18

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 41² - 18²

max_d = √2023 - 0 - 1681 - 324

max_d = √18

max_d = 4.2426406871193

Since max_d = 4.2426406871193 is not an integer, this is not a solution

b = 42

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 42²)

max_c = Floor(√2023 - 0 - 1764)

max_c = Floor(√259)

max_c = Floor(16.093476939431)

max_c = 16

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 42²)/2 = 129.5

When min_c = 12, then it is c² = 144 ≥ 129.5, so min_c = 12

c = 12

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 42² - 12²

max_d = √2023 - 0 - 1764 - 144

max_d = √115

max_d = 10.723805294764

Since max_d = 10.723805294764 is not an integer, this is not a solution

c = 13

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 42² - 13²

max_d = √2023 - 0 - 1764 - 169

max_d = √90

max_d = 9.4868329805051

Since max_d = 9.4868329805051 is not an integer, this is not a solution

c = 14

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 42² - 14²

max_d = √2023 - 0 - 1764 - 196

max_d = √63

max_d = 7.9372539331938

Since max_d = 7.9372539331938 is not an integer, this is not a solution

c = 15

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 42² - 15²

max_d = √2023 - 0 - 1764 - 225

max_d = √34

max_d = 5.8309518948453

Since max_d = 5.8309518948453 is not an integer, this is not a solution

c = 16

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 42² - 16²

max_d = √2023 - 0 - 1764 - 256

max_d = √3

max_d = 1.7320508075689

Since max_d = 1.7320508075689 is not an integer, this is not a solution

b = 43

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 43²)

max_c = Floor(√2023 - 0 - 1849)

max_c = Floor(√174)

max_c = Floor(13.190905958273)

max_c = 13

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 43²)/2 = 87

When min_c = 10, then it is c² = 100 ≥ 87, so min_c = 10

c = 10

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 43² - 10²

max_d = √2023 - 0 - 1849 - 100

max_d = √74

max_d = 8.6023252670426

Since max_d = 8.6023252670426 is not an integer, this is not a solution

c = 11

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 43² - 11²

max_d = √2023 - 0 - 1849 - 121

max_d = √53

max_d = 7.2801098892805

Since max_d = 7.2801098892805 is not an integer, this is not a solution

c = 12

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 43² - 12²

max_d = √2023 - 0 - 1849 - 144

max_d = √30

max_d = 5.4772255750517

Since max_d = 5.4772255750517 is not an integer, this is not a solution

c = 13

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 43² - 13²

max_d = √2023 - 0 - 1849 - 169

max_d = √5

max_d = 2.2360679774998

Since max_d = 2.2360679774998 is not an integer, this is not a solution

b = 44

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 0² - 44²)

max_c = Floor(√2023 - 0 - 1936)

max_c = Floor(√87)

max_c = Floor(9.3273790530888)

max_c = 9

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 0² - 44²)/2 = 43.5

When min_c = 7, then it is c² = 49 ≥ 43.5, so min_c = 7

c = 7

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 44² - 7²

max_d = √2023 - 0 - 1936 - 49

max_d = √38

max_d = 6.164414002969

Since max_d = 6.164414002969 is not an integer, this is not a solution

c = 8

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 44² - 8²

max_d = √2023 - 0 - 1936 - 64

max_d = √23

max_d = 4.7958315233127

Since max_d = 4.7958315233127 is not an integer, this is not a solution

c = 9

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 0² - 44² - 9²

max_d = √2023 - 0 - 1936 - 81

max_d = √6

max_d = 2.4494897427832

Since max_d = 2.4494897427832 is not an integer, this is not a solution

a = 1

Find max_b which is Floor(√n - a²)

max_b = Floor(√2023 - 1²)

max_b = Floor(√2023 - 1)

max_b = Floor(√2022)

max_b = Floor(44.966654311834)

max_b = 44

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 0 and increase by 1
Go until (n - a²)/3 → (2023 - 1²)/3 = 674

When min_b = 26, then it is b² = 676 ≥ 674, so min_b = 26

Test values for b in the range of (min_b, max_b)

(26, 44)

b = 26

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 26²)

max_c = Floor(√2023 - 1 - 676)

max_c = Floor(√1346)

max_c = Floor(36.687872655688)

max_c = 36

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 26²)/2 = 673

When min_c = 26, then it is c² = 676 ≥ 673, so min_c = 26

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 26² - 26²

max_d = √2023 - 1 - 676 - 676

max_d = √670

max_d = 25.88435821109

Since max_d = 25.88435821109 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 26² - 27²

max_d = √2023 - 1 - 676 - 729

max_d = √617

max_d = 24.839484696748

Since max_d = 24.839484696748 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 26² - 28²

max_d = √2023 - 1 - 676 - 784

max_d = √562

max_d = 23.706539182259

Since max_d = 23.706539182259 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 26² - 29²

max_d = √2023 - 1 - 676 - 841

max_d = √505

max_d = 22.472205054244

Since max_d = 22.472205054244 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 26² - 30²

max_d = √2023 - 1 - 676 - 900

max_d = √446

max_d = 21.118712081943

Since max_d = 21.118712081943 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 26² - 31²

max_d = √2023 - 1 - 676 - 961

max_d = √385

max_d = 19.621416870349

Since max_d = 19.621416870349 is not an integer, this is not a solution

c = 32

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 26² - 32²

max_d = √2023 - 1 - 676 - 1024

max_d = √322

max_d = 17.944358444926

Since max_d = 17.944358444926 is not an integer, this is not a solution

c = 33

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 26² - 33²

max_d = √2023 - 1 - 676 - 1089

max_d = √257

max_d = 16.031219541881

Since max_d = 16.031219541881 is not an integer, this is not a solution

c = 34

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 26² - 34²

max_d = √2023 - 1 - 676 - 1156

max_d = √190

max_d = 13.78404875209

Since max_d = 13.78404875209 is not an integer, this is not a solution

c = 35

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 26² - 35²

max_d = √2023 - 1 - 676 - 1225

max_d = √121

max_d = 11

Since max_d = 11, then (a, b, c, d) = (1, 26, 35, 11) is an integer solution proven below

1² + 26² + 35² + 11² → 1 + 676 + 1225 + 121 = 2023

c = 36

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 26² - 36²

max_d = √2023 - 1 - 676 - 1296

max_d = √50

max_d = 7.0710678118655

Since max_d = 7.0710678118655 is not an integer, this is not a solution

b = 27

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 27²)

max_c = Floor(√2023 - 1 - 729)

max_c = Floor(√1293)

max_c = Floor(35.95830919273)

max_c = 35

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 27²)/2 = 646.5

When min_c = 26, then it is c² = 676 ≥ 646.5, so min_c = 26

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 27² - 26²

max_d = √2023 - 1 - 729 - 676

max_d = √617

max_d = 24.839484696748

Since max_d = 24.839484696748 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 27² - 27²

max_d = √2023 - 1 - 729 - 729

max_d = √564

max_d = 23.748684174076

Since max_d = 23.748684174076 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 27² - 28²

max_d = √2023 - 1 - 729 - 784

max_d = √509

max_d = 22.561028345357

Since max_d = 22.561028345357 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 27² - 29²

max_d = √2023 - 1 - 729 - 841

max_d = √452

max_d = 21.260291625469

Since max_d = 21.260291625469 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 27² - 30²

max_d = √2023 - 1 - 729 - 900

max_d = √393

max_d = 19.824227601599

Since max_d = 19.824227601599 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 27² - 31²

max_d = √2023 - 1 - 729 - 961

max_d = √332

max_d = 18.220867158289

Since max_d = 18.220867158289 is not an integer, this is not a solution

c = 32

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 27² - 32²

max_d = √2023 - 1 - 729 - 1024

max_d = √269

max_d = 16.401219466857

Since max_d = 16.401219466857 is not an integer, this is not a solution

c = 33

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 27² - 33²

max_d = √2023 - 1 - 729 - 1089

max_d = √204

max_d = 14.282856857086

Since max_d = 14.282856857086 is not an integer, this is not a solution

c = 34

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 27² - 34²

max_d = √2023 - 1 - 729 - 1156

max_d = √137

max_d = 11.70469991072

Since max_d = 11.70469991072 is not an integer, this is not a solution

c = 35

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 27² - 35²

max_d = √2023 - 1 - 729 - 1225

max_d = √68

max_d = 8.2462112512353

Since max_d = 8.2462112512353 is not an integer, this is not a solution

b = 28

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 28²)

max_c = Floor(√2023 - 1 - 784)

max_c = Floor(√1238)

max_c = Floor(35.185224171518)

max_c = 35

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 28²)/2 = 619

When min_c = 25, then it is c² = 625 ≥ 619, so min_c = 25

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 28² - 25²

max_d = √2023 - 1 - 784 - 625

max_d = √613

max_d = 24.75883680628

Since max_d = 24.75883680628 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 28² - 26²

max_d = √2023 - 1 - 784 - 676

max_d = √562

max_d = 23.706539182259

Since max_d = 23.706539182259 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 28² - 27²

max_d = √2023 - 1 - 784 - 729

max_d = √509

max_d = 22.561028345357

Since max_d = 22.561028345357 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 28² - 28²

max_d = √2023 - 1 - 784 - 784

max_d = √454

max_d = 21.307275752663

Since max_d = 21.307275752663 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 28² - 29²

max_d = √2023 - 1 - 784 - 841

max_d = √397

max_d = 19.924858845171

Since max_d = 19.924858845171 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 28² - 30²

max_d = √2023 - 1 - 784 - 900

max_d = √338

max_d = 18.38477631085

Since max_d = 18.38477631085 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 28² - 31²

max_d = √2023 - 1 - 784 - 961

max_d = √277

max_d = 16.643316977093

Since max_d = 16.643316977093 is not an integer, this is not a solution

c = 32

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 28² - 32²

max_d = √2023 - 1 - 784 - 1024

max_d = √214

max_d = 14.628738838328

Since max_d = 14.628738838328 is not an integer, this is not a solution

c = 33

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 28² - 33²

max_d = √2023 - 1 - 784 - 1089

max_d = √149

max_d = 12.206555615734

Since max_d = 12.206555615734 is not an integer, this is not a solution

c = 34

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 28² - 34²

max_d = √2023 - 1 - 784 - 1156

max_d = √82

max_d = 9.0553851381374

Since max_d = 9.0553851381374 is not an integer, this is not a solution

c = 35

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 28² - 35²

max_d = √2023 - 1 - 784 - 1225

max_d = √13

max_d = 3.605551275464

Since max_d = 3.605551275464 is not an integer, this is not a solution

b = 29

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 29²)

max_c = Floor(√2023 - 1 - 841)

max_c = Floor(√1181)

max_c = Floor(34.365680554879)

max_c = 34

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 29²)/2 = 590.5

When min_c = 25, then it is c² = 625 ≥ 590.5, so min_c = 25

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 29² - 25²

max_d = √2023 - 1 - 841 - 625

max_d = √556

max_d = 23.579652245103

Since max_d = 23.579652245103 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 29² - 26²

max_d = √2023 - 1 - 841 - 676

max_d = √505

max_d = 22.472205054244

Since max_d = 22.472205054244 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 29² - 27²

max_d = √2023 - 1 - 841 - 729

max_d = √452

max_d = 21.260291625469

Since max_d = 21.260291625469 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 29² - 28²

max_d = √2023 - 1 - 841 - 784

max_d = √397

max_d = 19.924858845171

Since max_d = 19.924858845171 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 29² - 29²

max_d = √2023 - 1 - 841 - 841

max_d = √340

max_d = 18.439088914586

Since max_d = 18.439088914586 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 29² - 30²

max_d = √2023 - 1 - 841 - 900

max_d = √281

max_d = 16.76305461424

Since max_d = 16.76305461424 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 29² - 31²

max_d = √2023 - 1 - 841 - 961

max_d = √220

max_d = 14.832396974191

Since max_d = 14.832396974191 is not an integer, this is not a solution

c = 32

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 29² - 32²

max_d = √2023 - 1 - 841 - 1024

max_d = √157

max_d = 12.529964086142

Since max_d = 12.529964086142 is not an integer, this is not a solution

c = 33

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 29² - 33²

max_d = √2023 - 1 - 841 - 1089

max_d = √92

max_d = 9.5916630466254

Since max_d = 9.5916630466254 is not an integer, this is not a solution

c = 34

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 29² - 34²

max_d = √2023 - 1 - 841 - 1156

max_d = √25

max_d = 5

Since max_d = 5, then (a, b, c, d) = (1, 29, 34, 5) is an integer solution proven below

1² + 29² + 34² + 5² → 1 + 841 + 1156 + 25 = 2023

b = 30

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 30²)

max_c = Floor(√2023 - 1 - 900)

max_c = Floor(√1122)

max_c = Floor(33.496268448888)

max_c = 33

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 30²)/2 = 561

When min_c = 24, then it is c² = 576 ≥ 561, so min_c = 24

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 30² - 24²

max_d = √2023 - 1 - 900 - 576

max_d = √546

max_d = 23.366642891096

Since max_d = 23.366642891096 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 30² - 25²

max_d = √2023 - 1 - 900 - 625

max_d = √497

max_d = 22.293496809608

Since max_d = 22.293496809608 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 30² - 26²

max_d = √2023 - 1 - 900 - 676

max_d = √446

max_d = 21.118712081943

Since max_d = 21.118712081943 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 30² - 27²

max_d = √2023 - 1 - 900 - 729

max_d = √393

max_d = 19.824227601599

Since max_d = 19.824227601599 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 30² - 28²

max_d = √2023 - 1 - 900 - 784

max_d = √338

max_d = 18.38477631085

Since max_d = 18.38477631085 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 30² - 29²

max_d = √2023 - 1 - 900 - 841

max_d = √281

max_d = 16.76305461424

Since max_d = 16.76305461424 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 30² - 30²

max_d = √2023 - 1 - 900 - 900

max_d = √222

max_d = 14.899664425751

Since max_d = 14.899664425751 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 30² - 31²

max_d = √2023 - 1 - 900 - 961

max_d = √161

max_d = 12.68857754045

Since max_d = 12.68857754045 is not an integer, this is not a solution

c = 32

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 30² - 32²

max_d = √2023 - 1 - 900 - 1024

max_d = √98

max_d = 9.8994949366117

Since max_d = 9.8994949366117 is not an integer, this is not a solution

c = 33

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 30² - 33²

max_d = √2023 - 1 - 900 - 1089

max_d = √33

max_d = 5.744562646538

Since max_d = 5.744562646538 is not an integer, this is not a solution

b = 31

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 31²)

max_c = Floor(√2023 - 1 - 961)

max_c = Floor(√1061)

max_c = Floor(32.572994949805)

max_c = 32

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 31²)/2 = 530.5

When min_c = 24, then it is c² = 576 ≥ 530.5, so min_c = 24

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 31² - 24²

max_d = √2023 - 1 - 961 - 576

max_d = √485

max_d = 22.022715545545

Since max_d = 22.022715545545 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 31² - 25²

max_d = √2023 - 1 - 961 - 625

max_d = √436

max_d = 20.880613017821

Since max_d = 20.880613017821 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 31² - 26²

max_d = √2023 - 1 - 961 - 676

max_d = √385

max_d = 19.621416870349

Since max_d = 19.621416870349 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 31² - 27²

max_d = √2023 - 1 - 961 - 729

max_d = √332

max_d = 18.220867158289

Since max_d = 18.220867158289 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 31² - 28²

max_d = √2023 - 1 - 961 - 784

max_d = √277

max_d = 16.643316977093

Since max_d = 16.643316977093 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 31² - 29²

max_d = √2023 - 1 - 961 - 841

max_d = √220

max_d = 14.832396974191

Since max_d = 14.832396974191 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 31² - 30²

max_d = √2023 - 1 - 961 - 900

max_d = √161

max_d = 12.68857754045

Since max_d = 12.68857754045 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 31² - 31²

max_d = √2023 - 1 - 961 - 961

max_d = √100

max_d = 10

Since max_d = 10, then (a, b, c, d) = (1, 31, 31, 10) is an integer solution proven below

1² + 31² + 31² + 10² → 1 + 961 + 961 + 100 = 2023

c = 32

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 31² - 32²

max_d = √2023 - 1 - 961 - 1024

max_d = √37

max_d = 6.0827625302982

Since max_d = 6.0827625302982 is not an integer, this is not a solution

b = 32

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 32²)

max_c = Floor(√2023 - 1 - 1024)

max_c = Floor(√998)

max_c = Floor(31.591137997863)

max_c = 31

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 32²)/2 = 499

When min_c = 23, then it is c² = 529 ≥ 499, so min_c = 23

c = 23

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 32² - 23²

max_d = √2023 - 1 - 1024 - 529

max_d = √469

max_d = 21.656407827708

Since max_d = 21.656407827708 is not an integer, this is not a solution

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 32² - 24²

max_d = √2023 - 1 - 1024 - 576

max_d = √422

max_d = 20.542638584174

Since max_d = 20.542638584174 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 32² - 25²

max_d = √2023 - 1 - 1024 - 625

max_d = √373

max_d = 19.313207915828

Since max_d = 19.313207915828 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 32² - 26²

max_d = √2023 - 1 - 1024 - 676

max_d = √322

max_d = 17.944358444926

Since max_d = 17.944358444926 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 32² - 27²

max_d = √2023 - 1 - 1024 - 729

max_d = √269

max_d = 16.401219466857

Since max_d = 16.401219466857 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 32² - 28²

max_d = √2023 - 1 - 1024 - 784

max_d = √214

max_d = 14.628738838328

Since max_d = 14.628738838328 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 32² - 29²

max_d = √2023 - 1 - 1024 - 841

max_d = √157

max_d = 12.529964086142

Since max_d = 12.529964086142 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 32² - 30²

max_d = √2023 - 1 - 1024 - 900

max_d = √98

max_d = 9.8994949366117

Since max_d = 9.8994949366117 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 32² - 31²

max_d = √2023 - 1 - 1024 - 961

max_d = √37

max_d = 6.0827625302982

Since max_d = 6.0827625302982 is not an integer, this is not a solution

b = 33

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 33²)

max_c = Floor(√2023 - 1 - 1089)

max_c = Floor(√933)

max_c = Floor(30.545048698603)

max_c = 30

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 33²)/2 = 466.5

When min_c = 22, then it is c² = 484 ≥ 466.5, so min_c = 22

c = 22

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 33² - 22²

max_d = √2023 - 1 - 1089 - 484

max_d = √449

max_d = 21.189620100417

Since max_d = 21.189620100417 is not an integer, this is not a solution

c = 23

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 33² - 23²

max_d = √2023 - 1 - 1089 - 529

max_d = √404

max_d = 20.099751242242

Since max_d = 20.099751242242 is not an integer, this is not a solution

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 33² - 24²

max_d = √2023 - 1 - 1089 - 576

max_d = √357

max_d = 18.894443627691

Since max_d = 18.894443627691 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 33² - 25²

max_d = √2023 - 1 - 1089 - 625

max_d = √308

max_d = 17.549928774784

Since max_d = 17.549928774784 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 33² - 26²

max_d = √2023 - 1 - 1089 - 676

max_d = √257

max_d = 16.031219541881

Since max_d = 16.031219541881 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 33² - 27²

max_d = √2023 - 1 - 1089 - 729

max_d = √204

max_d = 14.282856857086

Since max_d = 14.282856857086 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 33² - 28²

max_d = √2023 - 1 - 1089 - 784

max_d = √149

max_d = 12.206555615734

Since max_d = 12.206555615734 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 33² - 29²

max_d = √2023 - 1 - 1089 - 841

max_d = √92

max_d = 9.5916630466254

Since max_d = 9.5916630466254 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 33² - 30²

max_d = √2023 - 1 - 1089 - 900

max_d = √33

max_d = 5.744562646538

Since max_d = 5.744562646538 is not an integer, this is not a solution

b = 34

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 34²)

max_c = Floor(√2023 - 1 - 1156)

max_c = Floor(√866)

max_c = Floor(29.427877939124)

max_c = 29

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 34²)/2 = 433

When min_c = 21, then it is c² = 441 ≥ 433, so min_c = 21

c = 21

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 34² - 21²

max_d = √2023 - 1 - 1156 - 441

max_d = √425

max_d = 20.615528128088

Since max_d = 20.615528128088 is not an integer, this is not a solution

c = 22

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 34² - 22²

max_d = √2023 - 1 - 1156 - 484

max_d = √382

max_d = 19.544820285692

Since max_d = 19.544820285692 is not an integer, this is not a solution

c = 23

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 34² - 23²

max_d = √2023 - 1 - 1156 - 529

max_d = √337

max_d = 18.357559750686

Since max_d = 18.357559750686 is not an integer, this is not a solution

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 34² - 24²

max_d = √2023 - 1 - 1156 - 576

max_d = √290

max_d = 17.029386365926

Since max_d = 17.029386365926 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 34² - 25²

max_d = √2023 - 1 - 1156 - 625

max_d = √241

max_d = 15.52417469626

Since max_d = 15.52417469626 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 34² - 26²

max_d = √2023 - 1 - 1156 - 676

max_d = √190

max_d = 13.78404875209

Since max_d = 13.78404875209 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 34² - 27²

max_d = √2023 - 1 - 1156 - 729

max_d = √137

max_d = 11.70469991072

Since max_d = 11.70469991072 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 34² - 28²

max_d = √2023 - 1 - 1156 - 784

max_d = √82

max_d = 9.0553851381374

Since max_d = 9.0553851381374 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 34² - 29²

max_d = √2023 - 1 - 1156 - 841

max_d = √25

max_d = 5

Since max_d = 5, then (a, b, c, d) = (1, 34, 29, 5) is an integer solution proven below

1² + 34² + 29² + 5² → 1 + 1156 + 841 + 25 = 2023

b = 35

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 35²)

max_c = Floor(√2023 - 1 - 1225)

max_c = Floor(√797)

max_c = Floor(28.231188426986)

max_c = 28

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 35²)/2 = 398.5

When min_c = 20, then it is c² = 400 ≥ 398.5, so min_c = 20

c = 20

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 35² - 20²

max_d = √2023 - 1 - 1225 - 400

max_d = √397

max_d = 19.924858845171

Since max_d = 19.924858845171 is not an integer, this is not a solution

c = 21

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 35² - 21²

max_d = √2023 - 1 - 1225 - 441

max_d = √356

max_d = 18.867962264113

Since max_d = 18.867962264113 is not an integer, this is not a solution

c = 22

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 35² - 22²

max_d = √2023 - 1 - 1225 - 484

max_d = √313

max_d = 17.691806012954

Since max_d = 17.691806012954 is not an integer, this is not a solution

c = 23

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 35² - 23²

max_d = √2023 - 1 - 1225 - 529

max_d = √268

max_d = 16.370705543745

Since max_d = 16.370705543745 is not an integer, this is not a solution

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 35² - 24²

max_d = √2023 - 1 - 1225 - 576

max_d = √221

max_d = 14.866068747319

Since max_d = 14.866068747319 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 35² - 25²

max_d = √2023 - 1 - 1225 - 625

max_d = √172

max_d = 13.114877048604

Since max_d = 13.114877048604 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 35² - 26²

max_d = √2023 - 1 - 1225 - 676

max_d = √121

max_d = 11

Since max_d = 11, then (a, b, c, d) = (1, 35, 26, 11) is an integer solution proven below

1² + 35² + 26² + 11² → 1 + 1225 + 676 + 121 = 2023

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 35² - 27²

max_d = √2023 - 1 - 1225 - 729

max_d = √68

max_d = 8.2462112512353

Since max_d = 8.2462112512353 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 35² - 28²

max_d = √2023 - 1 - 1225 - 784

max_d = √13

max_d = 3.605551275464

Since max_d = 3.605551275464 is not an integer, this is not a solution

b = 36

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 36²)

max_c = Floor(√2023 - 1 - 1296)

max_c = Floor(√726)

max_c = Floor(26.944387170615)

max_c = 26

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 36²)/2 = 363

When min_c = 20, then it is c² = 400 ≥ 363, so min_c = 20

c = 20

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 36² - 20²

max_d = √2023 - 1 - 1296 - 400

max_d = √326

max_d = 18.055470085268

Since max_d = 18.055470085268 is not an integer, this is not a solution

c = 21

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 36² - 21²

max_d = √2023 - 1 - 1296 - 441

max_d = √285

max_d = 16.881943016134

Since max_d = 16.881943016134 is not an integer, this is not a solution

c = 22

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 36² - 22²

max_d = √2023 - 1 - 1296 - 484

max_d = √242

max_d = 15.556349186104

Since max_d = 15.556349186104 is not an integer, this is not a solution

c = 23

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 36² - 23²

max_d = √2023 - 1 - 1296 - 529

max_d = √197

max_d = 14.035668847618

Since max_d = 14.035668847618 is not an integer, this is not a solution

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 36² - 24²

max_d = √2023 - 1 - 1296 - 576

max_d = √150

max_d = 12.247448713916

Since max_d = 12.247448713916 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 36² - 25²

max_d = √2023 - 1 - 1296 - 625

max_d = √101

max_d = 10.049875621121

Since max_d = 10.049875621121 is not an integer, this is not a solution

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 36² - 26²

max_d = √2023 - 1 - 1296 - 676

max_d = √50

max_d = 7.0710678118655

Since max_d = 7.0710678118655 is not an integer, this is not a solution

b = 37

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 37²)

max_c = Floor(√2023 - 1 - 1369)

max_c = Floor(√653)

max_c = Floor(25.553864678361)

max_c = 25

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 37²)/2 = 326.5

When min_c = 19, then it is c² = 361 ≥ 326.5, so min_c = 19

c = 19

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 37² - 19²

max_d = √2023 - 1 - 1369 - 361

max_d = √292

max_d = 17.088007490635

Since max_d = 17.088007490635 is not an integer, this is not a solution

c = 20

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 37² - 20²

max_d = √2023 - 1 - 1369 - 400

max_d = √253

max_d = 15.905973720587

Since max_d = 15.905973720587 is not an integer, this is not a solution

c = 21

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 37² - 21²

max_d = √2023 - 1 - 1369 - 441

max_d = √212

max_d = 14.560219778561

Since max_d = 14.560219778561 is not an integer, this is not a solution

c = 22

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 37² - 22²

max_d = √2023 - 1 - 1369 - 484

max_d = √169

max_d = 13

Since max_d = 13, then (a, b, c, d) = (1, 37, 22, 13) is an integer solution proven below

1² + 37² + 22² + 13² → 1 + 1369 + 484 + 169 = 2023

c = 23

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 37² - 23²

max_d = √2023 - 1 - 1369 - 529

max_d = √124

max_d = 11.13552872566

Since max_d = 11.13552872566 is not an integer, this is not a solution

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 37² - 24²

max_d = √2023 - 1 - 1369 - 576

max_d = √77

max_d = 8.7749643873921

Since max_d = 8.7749643873921 is not an integer, this is not a solution

c = 25

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 37² - 25²

max_d = √2023 - 1 - 1369 - 625

max_d = √28

max_d = 5.2915026221292

Since max_d = 5.2915026221292 is not an integer, this is not a solution

b = 38

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 38²)

max_c = Floor(√2023 - 1 - 1444)

max_c = Floor(√578)

max_c = Floor(24.041630560343)

max_c = 24

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 38²)/2 = 289

When min_c = 17, then it is c² = 289 ≥ 289, so min_c = 17

c = 17

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 38² - 17²

max_d = √2023 - 1 - 1444 - 289

max_d = √289

max_d = 17

Since max_d = 17, then (a, b, c, d) = (1, 38, 17, 17) is an integer solution proven below

1² + 38² + 17² + 17² → 1 + 1444 + 289 + 289 = 2023

c = 18

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 38² - 18²

max_d = √2023 - 1 - 1444 - 324

max_d = √254

max_d = 15.937377450509

Since max_d = 15.937377450509 is not an integer, this is not a solution

c = 19

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 38² - 19²

max_d = √2023 - 1 - 1444 - 361

max_d = √217

max_d = 14.730919862656

Since max_d = 14.730919862656 is not an integer, this is not a solution

c = 20

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 38² - 20²

max_d = √2023 - 1 - 1444 - 400

max_d = √178

max_d = 13.341664064126

Since max_d = 13.341664064126 is not an integer, this is not a solution

c = 21

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 38² - 21²

max_d = √2023 - 1 - 1444 - 441

max_d = √137

max_d = 11.70469991072

Since max_d = 11.70469991072 is not an integer, this is not a solution

c = 22

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 38² - 22²

max_d = √2023 - 1 - 1444 - 484

max_d = √94

max_d = 9.6953597148327

Since max_d = 9.6953597148327 is not an integer, this is not a solution

c = 23

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 38² - 23²

max_d = √2023 - 1 - 1444 - 529

max_d = √49

max_d = 7

Since max_d = 7, then (a, b, c, d) = (1, 38, 23, 7) is an integer solution proven below

1² + 38² + 23² + 7² → 1 + 1444 + 529 + 49 = 2023

c = 24

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 38² - 24²

max_d = √2023 - 1 - 1444 - 576

max_d = √2

max_d = 1.4142135623731

Since max_d = 1.4142135623731 is not an integer, this is not a solution

b = 39

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 39²)

max_c = Floor(√2023 - 1 - 1521)

max_c = Floor(√501)

max_c = Floor(22.383029285599)

max_c = 22

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 39²)/2 = 250.5

When min_c = 16, then it is c² = 256 ≥ 250.5, so min_c = 16

c = 16

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 39² - 16²

max_d = √2023 - 1 - 1521 - 256

max_d = √245

max_d = 15.652475842499

Since max_d = 15.652475842499 is not an integer, this is not a solution

c = 17

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 39² - 17²

max_d = √2023 - 1 - 1521 - 289

max_d = √212

max_d = 14.560219778561

Since max_d = 14.560219778561 is not an integer, this is not a solution

c = 18

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 39² - 18²

max_d = √2023 - 1 - 1521 - 324

max_d = √177

max_d = 13.30413469565

Since max_d = 13.30413469565 is not an integer, this is not a solution

c = 19

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 39² - 19²

max_d = √2023 - 1 - 1521 - 361

max_d = √140

max_d = 11.832159566199

Since max_d = 11.832159566199 is not an integer, this is not a solution

c = 20

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 39² - 20²

max_d = √2023 - 1 - 1521 - 400

max_d = √101

max_d = 10.049875621121

Since max_d = 10.049875621121 is not an integer, this is not a solution

c = 21

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 39² - 21²

max_d = √2023 - 1 - 1521 - 441

max_d = √60

max_d = 7.7459666924148

Since max_d = 7.7459666924148 is not an integer, this is not a solution

c = 22

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 39² - 22²

max_d = √2023 - 1 - 1521 - 484

max_d = √17

max_d = 4.1231056256177

Since max_d = 4.1231056256177 is not an integer, this is not a solution

b = 40

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 40²)

max_c = Floor(√2023 - 1 - 1600)

max_c = Floor(√422)

max_c = Floor(20.542638584174)

max_c = 20

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 40²)/2 = 211

When min_c = 15, then it is c² = 225 ≥ 211, so min_c = 15

c = 15

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 40² - 15²

max_d = √2023 - 1 - 1600 - 225

max_d = √197

max_d = 14.035668847618

Since max_d = 14.035668847618 is not an integer, this is not a solution

c = 16

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 40² - 16²

max_d = √2023 - 1 - 1600 - 256

max_d = √166

max_d = 12.884098726725

Since max_d = 12.884098726725 is not an integer, this is not a solution

c = 17

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 40² - 17²

max_d = √2023 - 1 - 1600 - 289

max_d = √133

max_d = 11.532562594671

Since max_d = 11.532562594671 is not an integer, this is not a solution

c = 18

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 40² - 18²

max_d = √2023 - 1 - 1600 - 324

max_d = √98

max_d = 9.8994949366117

Since max_d = 9.8994949366117 is not an integer, this is not a solution

c = 19

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 40² - 19²

max_d = √2023 - 1 - 1600 - 361

max_d = √61

max_d = 7.8102496759067

Since max_d = 7.8102496759067 is not an integer, this is not a solution

c = 20

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 40² - 20²

max_d = √2023 - 1 - 1600 - 400

max_d = √22

max_d = 4.6904157598234

Since max_d = 4.6904157598234 is not an integer, this is not a solution

b = 41

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 41²)

max_c = Floor(√2023 - 1 - 1681)

max_c = Floor(√341)

max_c = Floor(18.466185312619)

max_c = 18

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 41²)/2 = 170.5

When min_c = 14, then it is c² = 196 ≥ 170.5, so min_c = 14

c = 14

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 41² - 14²

max_d = √2023 - 1 - 1681 - 196

max_d = √145

max_d = 12.041594578792

Since max_d = 12.041594578792 is not an integer, this is not a solution

c = 15

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 41² - 15²

max_d = √2023 - 1 - 1681 - 225

max_d = √116

max_d = 10.770329614269

Since max_d = 10.770329614269 is not an integer, this is not a solution

c = 16

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 41² - 16²

max_d = √2023 - 1 - 1681 - 256

max_d = √85

max_d = 9.2195444572929

Since max_d = 9.2195444572929 is not an integer, this is not a solution

c = 17

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 41² - 17²

max_d = √2023 - 1 - 1681 - 289

max_d = √52

max_d = 7.211102550928

Since max_d = 7.211102550928 is not an integer, this is not a solution

c = 18

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 41² - 18²

max_d = √2023 - 1 - 1681 - 324

max_d = √17

max_d = 4.1231056256177

Since max_d = 4.1231056256177 is not an integer, this is not a solution

b = 42

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 42²)

max_c = Floor(√2023 - 1 - 1764)

max_c = Floor(√258)

max_c = Floor(16.062378404209)

max_c = 16

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 42²)/2 = 129

When min_c = 12, then it is c² = 144 ≥ 129, so min_c = 12

c = 12

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 42² - 12²

max_d = √2023 - 1 - 1764 - 144

max_d = √114

max_d = 10.677078252031

Since max_d = 10.677078252031 is not an integer, this is not a solution

c = 13

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 42² - 13²

max_d = √2023 - 1 - 1764 - 169

max_d = √89

max_d = 9.4339811320566

Since max_d = 9.4339811320566 is not an integer, this is not a solution

c = 14

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 42² - 14²

max_d = √2023 - 1 - 1764 - 196

max_d = √62

max_d = 7.8740078740118

Since max_d = 7.8740078740118 is not an integer, this is not a solution

c = 15

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 42² - 15²

max_d = √2023 - 1 - 1764 - 225

max_d = √33

max_d = 5.744562646538

Since max_d = 5.744562646538 is not an integer, this is not a solution

c = 16

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 42² - 16²

max_d = √2023 - 1 - 1764 - 256

max_d = √2

max_d = 1.4142135623731

Since max_d = 1.4142135623731 is not an integer, this is not a solution

b = 43

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 43²)

max_c = Floor(√2023 - 1 - 1849)

max_c = Floor(√173)

max_c = Floor(13.152946437966)

max_c = 13

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 43²)/2 = 86.5

When min_c = 10, then it is c² = 100 ≥ 86.5, so min_c = 10

c = 10

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 43² - 10²

max_d = √2023 - 1 - 1849 - 100

max_d = √73

max_d = 8.5440037453175

Since max_d = 8.5440037453175 is not an integer, this is not a solution

c = 11

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 43² - 11²

max_d = √2023 - 1 - 1849 - 121

max_d = √52

max_d = 7.211102550928

Since max_d = 7.211102550928 is not an integer, this is not a solution

c = 12

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 43² - 12²

max_d = √2023 - 1 - 1849 - 144

max_d = √29

max_d = 5.3851648071345

Since max_d = 5.3851648071345 is not an integer, this is not a solution

c = 13

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 43² - 13²

max_d = √2023 - 1 - 1849 - 169

max_d = √4

max_d = 2

Since max_d = 2, then (a, b, c, d) = (1, 43, 13, 2) is an integer solution proven below

1² + 43² + 13² + 2² → 1 + 1849 + 169 + 4 = 2023

b = 44

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 1² - 44²)

max_c = Floor(√2023 - 1 - 1936)

max_c = Floor(√86)

max_c = Floor(9.2736184954957)

max_c = 9

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 1² - 44²)/2 = 43

When min_c = 7, then it is c² = 49 ≥ 43, so min_c = 7

c = 7

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 44² - 7²

max_d = √2023 - 1 - 1936 - 49

max_d = √37

max_d = 6.0827625302982

Since max_d = 6.0827625302982 is not an integer, this is not a solution

c = 8

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 44² - 8²

max_d = √2023 - 1 - 1936 - 64

max_d = √22

max_d = 4.6904157598234

Since max_d = 4.6904157598234 is not an integer, this is not a solution

c = 9

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 1² - 44² - 9²

max_d = √2023 - 1 - 1936 - 81

max_d = √5

max_d = 2.2360679774998

Since max_d = 2.2360679774998 is not an integer, this is not a solution

a = 2

Find max_b which is Floor(√n - a²)

max_b = Floor(√2023 - 2²)

max_b = Floor(√2023 - 4)

max_b = Floor(√2019)

max_b = Floor(44.933283877322)

max_b = 44

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 0 and increase by 1
Go until (n - a²)/3 → (2023 - 2²)/3 = 673

When min_b = 26, then it is b² = 676 ≥ 673, so min_b = 26

Test values for b in the range of (min_b, max_b)

(26, 44)

b = 26

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 2² - 26²)

max_c = Floor(√2023 - 4 - 676)

max_c = Floor(√1343)

max_c = Floor(36.646964403617)

max_c = 36

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 2² - 26²)/2 = 671.5

When min_c = 26, then it is c² = 676 ≥ 671.5, so min_c = 26

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 26² - 26²

max_d = √2023 - 4 - 676 - 676

max_d = √667

max_d = 25.82634314029

Since max_d = 25.82634314029 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 26² - 27²

max_d = √2023 - 4 - 676 - 729

max_d = √614

max_d = 24.779023386728

Since max_d = 24.779023386728 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 26² - 28²

max_d = √2023 - 4 - 676 - 784

max_d = √559

max_d = 23.643180835074

Since max_d = 23.643180835074 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 26² - 29²

max_d = √2023 - 4 - 676 - 841

max_d = √502

max_d = 22.405356502408

Since max_d = 22.405356502408 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 26² - 30²

max_d = √2023 - 4 - 676 - 900

max_d = √443

max_d = 21.047565179849

Since max_d = 21.047565179849 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 26² - 31²

max_d = √2023 - 4 - 676 - 961

max_d = √382

max_d = 19.544820285692

Since max_d = 19.544820285692 is not an integer, this is not a solution

c = 32

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 26² - 32²

max_d = √2023 - 4 - 676 - 1024

max_d = √319

max_d = 17.860571099492

Since max_d = 17.860571099492 is not an integer, this is not a solution

c = 33

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 26² - 33²

max_d = √2023 - 4 - 676 - 1089

max_d = √254

max_d = 15.937377450509

Since max_d = 15.937377450509 is not an integer, this is not a solution

c = 34

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 26² - 34²

max_d = √2023 - 4 - 676 - 1156

max_d = √187

max_d = 13.674794331177

Since max_d = 13.674794331177 is not an integer, this is not a solution

c = 35

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 26² - 35²

max_d = √2023 - 4 - 676 - 1225

max_d = √118

max_d = 10.8627804912

Since max_d = 10.8627804912 is not an integer, this is not a solution

c = 36

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 26² - 36²

max_d = √2023 - 4 - 676 - 1296

max_d = √47

max_d = 6.855654600401

Since max_d = 6.855654600401 is not an integer, this is not a solution

b = 27

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√2023 - 2² - 27²)

max_c = Floor(√2023 - 4 - 729)

max_c = Floor(√1290)

max_c = Floor(35.916569992136)

max_c = 35

Step 5b. Obtain the first value of b such that b² ≥ (n - a²)/3

Call it min_b

Start with min_c = 0 and increase by 1

Go until (n - a² - b² )/2 → (2023 - 2² - 27²)/2 = 645

When min_c = 26, then it is c² = 676 ≥ 645, so min_c = 26

c = 26

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 27² - 26²

max_d = √2023 - 4 - 729 - 676

max_d = √614

max_d = 24.779023386728

Since max_d = 24.779023386728 is not an integer, this is not a solution

c = 27

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 27² - 27²

max_d = √2023 - 4 - 729 - 729

max_d = √561

max_d = 23.685438564654

Since max_d = 23.685438564654 is not an integer, this is not a solution

c = 28

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 27² - 28²

max_d = √2023 - 4 - 729 - 784

max_d = √506

max_d = 22.494443758404

Since max_d = 22.494443758404 is not an integer, this is not a solution

c = 29

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 27² - 29²

max_d = √2023 - 4 - 729 - 841

max_d = √449

max_d = 21.189620100417

Since max_d = 21.189620100417 is not an integer, this is not a solution

c = 30

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 27² - 30²

max_d = √2023 - 4 - 729 - 900

max_d = √390

max_d = 19.748417658131

Since max_d = 19.748417658131 is not an integer, this is not a solution

c = 31

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 27² - 31²

max_d = √2023 - 4 - 729 - 961

max_d = √329

max_d = 18.138357147217

Since max_d = 18.138357147217 is not an integer, this is not a solution

c = 32

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 27² - 32²

max_d = √2023 - 4 - 729 - 1024

max_d = √266

max_d = 16.3095064303

Since max_d = 16.3095064303 is not an integer, this is not a solution

c = 33

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 27² - 33²

max_d = √2023 - 4 - 729 - 1089

max_d = √201

max_d = 14.177446878758

Since max_d = 14.177446878758 is not an integer, this is not a solution

c = 34

See if d is an integer solution which is √n - a² - b²

max_d = √2023 - 2² - 27² - 34²