Lagrange four-square (Bachet Conjecture) for 454

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Show the Lagrange Four Square Theorem for

454

Lagrange Four Square Definition

For any natural number (p), we write as

p = a² + b² + c² + d²

Determine max_a:

Floor(√454) = Floor(21.307275752663)

Floor(21.307275752663) = 21
This is called max_a

Determine min_a:

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

Start with min_a = 1 and increase by 1

Continue until we reach or breach n/4 → 454/4 = 113.5

When min_a = 11, then it is a² = 121 ≥ 113.5, so min_a = 11

Find a in the range of (min_a, max_a)

(11, 21)

a = 11

Find max_b which is Floor(√n - a²)

max_b = Floor(√454 - 11²)

max_b = Floor(√454 - 121)

max_b = Floor(√333)

max_b = Floor(18.248287590895)

max_b = 18

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 1 and increase by 1
Go until (n - a²)/3 → (454 - 11²)/3 = 111

When min_b = 11, then it is b² = 121 ≥ 111, so min_b = 11

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

(11, 18)

b = 11

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 11² - 11²)

max_c = Floor(√454 - 121 - 121)

max_c = Floor(√212)

max_c = Floor(14.560219778561)

max_c = 14

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 11² - 11²)/2 = 106

When min_c = 11, then it is c² = 121 ≥ 106, so min_c = 11

c = 11

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

max_d = √454 - 11² - 11² - 11²

max_d = √454 - 121 - 121 - 121

max_d = √91

max_d = 9.5393920141695

Since max_d = 9.5393920141695 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 = √454 - 11² - 11² - 12²

max_d = √454 - 121 - 121 - 144

max_d = √68

max_d = 8.2462112512353

Since max_d = 8.2462112512353 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 = √454 - 11² - 11² - 13²

max_d = √454 - 121 - 121 - 169

max_d = √43

max_d = 6.557438524302

Since max_d = 6.557438524302 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 = √454 - 11² - 11² - 14²

max_d = √454 - 121 - 121 - 196

max_d = √16

max_d = 4

Since max_d = 4, then (a, b, c, d) = (11, 11, 14, 4) is an integer solution proven below

11² + 11² + 14² + 4² → 121 + 121 + 196 + 16 = 454

b = 12

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 11² - 12²)

max_c = Floor(√454 - 121 - 144)

max_c = Floor(√189)

max_c = Floor(13.747727084868)

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 = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 11² - 12²)/2 = 94.5

When min_c = 10, then it is c² = 100 ≥ 94.5, so min_c = 10

c = 10

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

max_d = √454 - 11² - 12² - 10²

max_d = √454 - 121 - 144 - 100

max_d = √89

max_d = 9.4339811320566

Since max_d = 9.4339811320566 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 = √454 - 11² - 12² - 11²

max_d = √454 - 121 - 144 - 121

max_d = √68

max_d = 8.2462112512353

Since max_d = 8.2462112512353 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 = √454 - 11² - 12² - 12²

max_d = √454 - 121 - 144 - 144

max_d = √45

max_d = 6.7082039324994

Since max_d = 6.7082039324994 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 = √454 - 11² - 12² - 13²

max_d = √454 - 121 - 144 - 169

max_d = √20

max_d = 4.4721359549996

Since max_d = 4.4721359549996 is not an integer, this is not a solution

b = 13

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 11² - 13²)

max_c = Floor(√454 - 121 - 169)

max_c = Floor(√164)

max_c = Floor(12.806248474866)

max_c = 12

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 11² - 13²)/2 = 82

When min_c = 10, then it is c² = 100 ≥ 82, so min_c = 10

c = 10

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

max_d = √454 - 11² - 13² - 10²

max_d = √454 - 121 - 169 - 100

max_d = √64

max_d = 8

Since max_d = 8, then (a, b, c, d) = (11, 13, 10, 8) is an integer solution proven below

11² + 13² + 10² + 8² → 121 + 169 + 100 + 64 = 454

c = 11

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

max_d = √454 - 11² - 13² - 11²

max_d = √454 - 121 - 169 - 121

max_d = √43

max_d = 6.557438524302

Since max_d = 6.557438524302 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 = √454 - 11² - 13² - 12²

max_d = √454 - 121 - 169 - 144

max_d = √20

max_d = 4.4721359549996

Since max_d = 4.4721359549996 is not an integer, this is not a solution

b = 14

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 11² - 14²)

max_c = Floor(√454 - 121 - 196)

max_c = Floor(√137)

max_c = Floor(11.70469991072)

max_c = 11

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 11² - 14²)/2 = 68.5

When min_c = 9, then it is c² = 81 ≥ 68.5, so min_c = 9

c = 9

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

max_d = √454 - 11² - 14² - 9²

max_d = √454 - 121 - 196 - 81

max_d = √56

max_d = 7.4833147735479

Since max_d = 7.4833147735479 is not an integer, this is not a solution

c = 10

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

max_d = √454 - 11² - 14² - 10²

max_d = √454 - 121 - 196 - 100

max_d = √37

max_d = 6.0827625302982

Since max_d = 6.0827625302982 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 = √454 - 11² - 14² - 11²

max_d = √454 - 121 - 196 - 121

max_d = √16

max_d = 4

Since max_d = 4, then (a, b, c, d) = (11, 14, 11, 4) is an integer solution proven below

11² + 14² + 11² + 4² → 121 + 196 + 121 + 16 = 454

b = 15

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 11² - 15²)

max_c = Floor(√454 - 121 - 225)

max_c = Floor(√108)

max_c = Floor(10.392304845413)

max_c = 10

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 11² - 15²)/2 = 54

When min_c = 8, then it is c² = 64 ≥ 54, so min_c = 8

c = 8

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

max_d = √454 - 11² - 15² - 8²

max_d = √454 - 121 - 225 - 64

max_d = √44

max_d = 6.6332495807108

Since max_d = 6.6332495807108 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 = √454 - 11² - 15² - 9²

max_d = √454 - 121 - 225 - 81

max_d = √27

max_d = 5.1961524227066

Since max_d = 5.1961524227066 is not an integer, this is not a solution

c = 10

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

max_d = √454 - 11² - 15² - 10²

max_d = √454 - 121 - 225 - 100

max_d = √8

max_d = 2.8284271247462

Since max_d = 2.8284271247462 is not an integer, this is not a solution

b = 16

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 11² - 16²)

max_c = Floor(√454 - 121 - 256)

max_c = Floor(√77)

max_c = Floor(8.7749643873921)

max_c = 8

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 11² - 16²)/2 = 38.5

When min_c = 7, then it is c² = 49 ≥ 38.5, so min_c = 7

c = 7

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

max_d = √454 - 11² - 16² - 7²

max_d = √454 - 121 - 256 - 49

max_d = √28

max_d = 5.2915026221292

Since max_d = 5.2915026221292 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 = √454 - 11² - 16² - 8²

max_d = √454 - 121 - 256 - 64

max_d = √13

max_d = 3.605551275464

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

b = 17

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 11² - 17²)

max_c = Floor(√454 - 121 - 289)

max_c = Floor(√44)

max_c = Floor(6.6332495807108)

max_c = 6

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 11² - 17²)/2 = 22

When min_c = 5, then it is c² = 25 ≥ 22, so min_c = 5

c = 5

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

max_d = √454 - 11² - 17² - 5²

max_d = √454 - 121 - 289 - 25

max_d = √19

max_d = 4.3588989435407

Since max_d = 4.3588989435407 is not an integer, this is not a solution

c = 6

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

max_d = √454 - 11² - 17² - 6²

max_d = √454 - 121 - 289 - 36

max_d = √8

max_d = 2.8284271247462

Since max_d = 2.8284271247462 is not an integer, this is not a solution

b = 18

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 11² - 18²)

max_c = Floor(√454 - 121 - 324)

max_c = Floor(√9)

max_c = Floor(3)

max_c = 3

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 11² - 18²)/2 = 4.5

When min_c = 3, then it is c² = 9 ≥ 4.5, so min_c = 3

c = 3

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

max_d = √454 - 11² - 18² - 3²

max_d = √454 - 121 - 324 - 9

max_d = √0

max_d = 0

Since max_d = 0, then (a, b, c, d) = (11, 18, 3, 0) is an integer solution proven below

11² + 18² + 3² + 0² → 121 + 324 + 9 + 0 = 454

a = 12

Find max_b which is Floor(√n - a²)

max_b = Floor(√454 - 12²)

max_b = Floor(√454 - 144)

max_b = Floor(√310)

max_b = Floor(17.606816861659)

max_b = 17

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 1 and increase by 1
Go until (n - a²)/3 → (454 - 12²)/3 = 103.33333333333

When min_b = 11, then it is b² = 121 ≥ 103.33333333333, so min_b = 11

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

(11, 17)

b = 11

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 12² - 11²)

max_c = Floor(√454 - 144 - 121)

max_c = Floor(√189)

max_c = Floor(13.747727084868)

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 = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 12² - 11²)/2 = 94.5

When min_c = 10, then it is c² = 100 ≥ 94.5, so min_c = 10

c = 10

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

max_d = √454 - 12² - 11² - 10²

max_d = √454 - 144 - 121 - 100

max_d = √89

max_d = 9.4339811320566

Since max_d = 9.4339811320566 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 = √454 - 12² - 11² - 11²

max_d = √454 - 144 - 121 - 121

max_d = √68

max_d = 8.2462112512353

Since max_d = 8.2462112512353 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 = √454 - 12² - 11² - 12²

max_d = √454 - 144 - 121 - 144

max_d = √45

max_d = 6.7082039324994

Since max_d = 6.7082039324994 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 = √454 - 12² - 11² - 13²

max_d = √454 - 144 - 121 - 169

max_d = √20

max_d = 4.4721359549996

Since max_d = 4.4721359549996 is not an integer, this is not a solution

b = 12

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 12² - 12²)

max_c = Floor(√454 - 144 - 144)

max_c = Floor(√166)

max_c = Floor(12.884098726725)

max_c = 12

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 12² - 12²)/2 = 83

When min_c = 10, then it is c² = 100 ≥ 83, so min_c = 10

c = 10

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

max_d = √454 - 12² - 12² - 10²

max_d = √454 - 144 - 144 - 100

max_d = √66

max_d = 8.124038404636

Since max_d = 8.124038404636 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 = √454 - 12² - 12² - 11²

max_d = √454 - 144 - 144 - 121

max_d = √45

max_d = 6.7082039324994

Since max_d = 6.7082039324994 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 = √454 - 12² - 12² - 12²

max_d = √454 - 144 - 144 - 144

max_d = √22

max_d = 4.6904157598234

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

b = 13

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 12² - 13²)

max_c = Floor(√454 - 144 - 169)

max_c = Floor(√141)

max_c = Floor(11.874342087038)

max_c = 11

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 12² - 13²)/2 = 70.5

When min_c = 9, then it is c² = 81 ≥ 70.5, so min_c = 9

c = 9

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

max_d = √454 - 12² - 13² - 9²

max_d = √454 - 144 - 169 - 81

max_d = √60

max_d = 7.7459666924148

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

c = 10

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

max_d = √454 - 12² - 13² - 10²

max_d = √454 - 144 - 169 - 100

max_d = √41

max_d = 6.4031242374328

Since max_d = 6.4031242374328 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 = √454 - 12² - 13² - 11²

max_d = √454 - 144 - 169 - 121

max_d = √20

max_d = 4.4721359549996

Since max_d = 4.4721359549996 is not an integer, this is not a solution

b = 14

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 12² - 14²)

max_c = Floor(√454 - 144 - 196)

max_c = Floor(√114)

max_c = Floor(10.677078252031)

max_c = 10

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 12² - 14²)/2 = 57

When min_c = 8, then it is c² = 64 ≥ 57, so min_c = 8

c = 8

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

max_d = √454 - 12² - 14² - 8²

max_d = √454 - 144 - 196 - 64

max_d = √50

max_d = 7.0710678118655

Since max_d = 7.0710678118655 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 = √454 - 12² - 14² - 9²

max_d = √454 - 144 - 196 - 81

max_d = √33

max_d = 5.744562646538

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

c = 10

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

max_d = √454 - 12² - 14² - 10²

max_d = √454 - 144 - 196 - 100

max_d = √14

max_d = 3.7416573867739

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

b = 15

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 12² - 15²)

max_c = Floor(√454 - 144 - 225)

max_c = Floor(√85)

max_c = Floor(9.2195444572929)

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 = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 12² - 15²)/2 = 42.5

When min_c = 7, then it is c² = 49 ≥ 42.5, so min_c = 7

c = 7

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

max_d = √454 - 12² - 15² - 7²

max_d = √454 - 144 - 225 - 49

max_d = √36

max_d = 6

Since max_d = 6, then (a, b, c, d) = (12, 15, 7, 6) is an integer solution proven below

12² + 15² + 7² + 6² → 144 + 225 + 49 + 36 = 454

c = 8

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

max_d = √454 - 12² - 15² - 8²

max_d = √454 - 144 - 225 - 64

max_d = √21

max_d = 4.5825756949558

Since max_d = 4.5825756949558 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 = √454 - 12² - 15² - 9²

max_d = √454 - 144 - 225 - 81

max_d = √4

max_d = 2

Since max_d = 2, then (a, b, c, d) = (12, 15, 9, 2) is an integer solution proven below

12² + 15² + 9² + 2² → 144 + 225 + 81 + 4 = 454

b = 16

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 12² - 16²)

max_c = Floor(√454 - 144 - 256)

max_c = Floor(√54)

max_c = Floor(7.3484692283495)

max_c = 7

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 12² - 16²)/2 = 27

When min_c = 6, then it is c² = 36 ≥ 27, so min_c = 6

c = 6

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

max_d = √454 - 12² - 16² - 6²

max_d = √454 - 144 - 256 - 36

max_d = √18

max_d = 4.2426406871193

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

c = 7

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

max_d = √454 - 12² - 16² - 7²

max_d = √454 - 144 - 256 - 49

max_d = √5

max_d = 2.2360679774998

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

b = 17

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 12² - 17²)

max_c = Floor(√454 - 144 - 289)

max_c = Floor(√21)

max_c = Floor(4.5825756949558)

max_c = 4

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 12² - 17²)/2 = 10.5

When min_c = 4, then it is c² = 16 ≥ 10.5, so min_c = 4

c = 4

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

max_d = √454 - 12² - 17² - 4²

max_d = √454 - 144 - 289 - 16

max_d = √5

max_d = 2.2360679774998

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

a = 13

Find max_b which is Floor(√n - a²)

max_b = Floor(√454 - 13²)

max_b = Floor(√454 - 169)

max_b = Floor(√285)

max_b = Floor(16.881943016134)

max_b = 16

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 1 and increase by 1
Go until (n - a²)/3 → (454 - 13²)/3 = 95

When min_b = 10, then it is b² = 100 ≥ 95, so min_b = 10

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

(10, 16)

b = 10

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 13² - 10²)

max_c = Floor(√454 - 169 - 100)

max_c = Floor(√185)

max_c = Floor(13.601470508735)

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 = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 13² - 10²)/2 = 92.5

When min_c = 10, then it is c² = 100 ≥ 92.5, so min_c = 10

c = 10

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

max_d = √454 - 13² - 10² - 10²

max_d = √454 - 169 - 100 - 100

max_d = √85

max_d = 9.2195444572929

Since max_d = 9.2195444572929 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 = √454 - 13² - 10² - 11²

max_d = √454 - 169 - 100 - 121

max_d = √64

max_d = 8

Since max_d = 8, then (a, b, c, d) = (13, 10, 11, 8) is an integer solution proven below

13² + 10² + 11² + 8² → 169 + 100 + 121 + 64 = 454

c = 12

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

max_d = √454 - 13² - 10² - 12²

max_d = √454 - 169 - 100 - 144

max_d = √41

max_d = 6.4031242374328

Since max_d = 6.4031242374328 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 = √454 - 13² - 10² - 13²

max_d = √454 - 169 - 100 - 169

max_d = √16

max_d = 4

Since max_d = 4, then (a, b, c, d) = (13, 10, 13, 4) is an integer solution proven below

13² + 10² + 13² + 4² → 169 + 100 + 169 + 16 = 454

b = 11

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 13² - 11²)

max_c = Floor(√454 - 169 - 121)

max_c = Floor(√164)

max_c = Floor(12.806248474866)

max_c = 12

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 13² - 11²)/2 = 82

When min_c = 10, then it is c² = 100 ≥ 82, so min_c = 10

c = 10

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

max_d = √454 - 13² - 11² - 10²

max_d = √454 - 169 - 121 - 100

max_d = √64

max_d = 8

Since max_d = 8, then (a, b, c, d) = (13, 11, 10, 8) is an integer solution proven below

13² + 11² + 10² + 8² → 169 + 121 + 100 + 64 = 454

c = 11

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

max_d = √454 - 13² - 11² - 11²

max_d = √454 - 169 - 121 - 121

max_d = √43

max_d = 6.557438524302

Since max_d = 6.557438524302 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 = √454 - 13² - 11² - 12²

max_d = √454 - 169 - 121 - 144

max_d = √20

max_d = 4.4721359549996

Since max_d = 4.4721359549996 is not an integer, this is not a solution

b = 12

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 13² - 12²)

max_c = Floor(√454 - 169 - 144)

max_c = Floor(√141)

max_c = Floor(11.874342087038)

max_c = 11

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 13² - 12²)/2 = 70.5

When min_c = 9, then it is c² = 81 ≥ 70.5, so min_c = 9

c = 9

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

max_d = √454 - 13² - 12² - 9²

max_d = √454 - 169 - 144 - 81

max_d = √60

max_d = 7.7459666924148

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

c = 10

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

max_d = √454 - 13² - 12² - 10²

max_d = √454 - 169 - 144 - 100

max_d = √41

max_d = 6.4031242374328

Since max_d = 6.4031242374328 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 = √454 - 13² - 12² - 11²

max_d = √454 - 169 - 144 - 121

max_d = √20

max_d = 4.4721359549996

Since max_d = 4.4721359549996 is not an integer, this is not a solution

b = 13

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 13² - 13²)

max_c = Floor(√454 - 169 - 169)

max_c = Floor(√116)

max_c = Floor(10.770329614269)

max_c = 10

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 13² - 13²)/2 = 58

When min_c = 8, then it is c² = 64 ≥ 58, so min_c = 8

c = 8

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

max_d = √454 - 13² - 13² - 8²

max_d = √454 - 169 - 169 - 64

max_d = √52

max_d = 7.211102550928

Since max_d = 7.211102550928 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 = √454 - 13² - 13² - 9²

max_d = √454 - 169 - 169 - 81

max_d = √35

max_d = 5.9160797830996

Since max_d = 5.9160797830996 is not an integer, this is not a solution

c = 10

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

max_d = √454 - 13² - 13² - 10²

max_d = √454 - 169 - 169 - 100

max_d = √16

max_d = 4

Since max_d = 4, then (a, b, c, d) = (13, 13, 10, 4) is an integer solution proven below

13² + 13² + 10² + 4² → 169 + 169 + 100 + 16 = 454

b = 14

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 13² - 14²)

max_c = Floor(√454 - 169 - 196)

max_c = Floor(√89)

max_c = Floor(9.4339811320566)

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 = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 13² - 14²)/2 = 44.5

When min_c = 7, then it is c² = 49 ≥ 44.5, so min_c = 7

c = 7

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

max_d = √454 - 13² - 14² - 7²

max_d = √454 - 169 - 196 - 49

max_d = √40

max_d = 6.3245553203368

Since max_d = 6.3245553203368 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 = √454 - 13² - 14² - 8²

max_d = √454 - 169 - 196 - 64

max_d = √25

max_d = 5

Since max_d = 5, then (a, b, c, d) = (13, 14, 8, 5) is an integer solution proven below

13² + 14² + 8² + 5² → 169 + 196 + 64 + 25 = 454

c = 9

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

max_d = √454 - 13² - 14² - 9²

max_d = √454 - 169 - 196 - 81

max_d = √8

max_d = 2.8284271247462

Since max_d = 2.8284271247462 is not an integer, this is not a solution

b = 15

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 13² - 15²)

max_c = Floor(√454 - 169 - 225)

max_c = Floor(√60)

max_c = Floor(7.7459666924148)

max_c = 7

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 13² - 15²)/2 = 30

When min_c = 6, then it is c² = 36 ≥ 30, so min_c = 6

c = 6

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

max_d = √454 - 13² - 15² - 6²

max_d = √454 - 169 - 225 - 36

max_d = √24

max_d = 4.8989794855664

Since max_d = 4.8989794855664 is not an integer, this is not a solution

c = 7

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

max_d = √454 - 13² - 15² - 7²

max_d = √454 - 169 - 225 - 49

max_d = √11

max_d = 3.3166247903554

Since max_d = 3.3166247903554 is not an integer, this is not a solution

b = 16

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 13² - 16²)

max_c = Floor(√454 - 169 - 256)

max_c = Floor(√29)

max_c = Floor(5.3851648071345)

max_c = 5

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 13² - 16²)/2 = 14.5

When min_c = 4, then it is c² = 16 ≥ 14.5, so min_c = 4

c = 4

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

max_d = √454 - 13² - 16² - 4²

max_d = √454 - 169 - 256 - 16

max_d = √13

max_d = 3.605551275464

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

c = 5

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

max_d = √454 - 13² - 16² - 5²

max_d = √454 - 169 - 256 - 25

max_d = √4

max_d = 2

Since max_d = 2, then (a, b, c, d) = (13, 16, 5, 2) is an integer solution proven below

13² + 16² + 5² + 2² → 169 + 256 + 25 + 4 = 454

a = 14

Find max_b which is Floor(√n - a²)

max_b = Floor(√454 - 14²)

max_b = Floor(√454 - 196)

max_b = Floor(√258)

max_b = Floor(16.062378404209)

max_b = 16

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 1 and increase by 1
Go until (n - a²)/3 → (454 - 14²)/3 = 86

When min_b = 10, then it is b² = 100 ≥ 86, so min_b = 10

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

(10, 16)

b = 10

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 14² - 10²)

max_c = Floor(√454 - 196 - 100)

max_c = Floor(√158)

max_c = Floor(12.569805089977)

max_c = 12

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 14² - 10²)/2 = 79

When min_c = 9, then it is c² = 81 ≥ 79, so min_c = 9

c = 9

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

max_d = √454 - 14² - 10² - 9²

max_d = √454 - 196 - 100 - 81

max_d = √77

max_d = 8.7749643873921

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

c = 10

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

max_d = √454 - 14² - 10² - 10²

max_d = √454 - 196 - 100 - 100

max_d = √58

max_d = 7.6157731058639

Since max_d = 7.6157731058639 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 = √454 - 14² - 10² - 11²

max_d = √454 - 196 - 100 - 121

max_d = √37

max_d = 6.0827625302982

Since max_d = 6.0827625302982 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 = √454 - 14² - 10² - 12²

max_d = √454 - 196 - 100 - 144

max_d = √14

max_d = 3.7416573867739

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

b = 11

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 14² - 11²)

max_c = Floor(√454 - 196 - 121)

max_c = Floor(√137)

max_c = Floor(11.70469991072)

max_c = 11

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 14² - 11²)/2 = 68.5

When min_c = 9, then it is c² = 81 ≥ 68.5, so min_c = 9

c = 9

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

max_d = √454 - 14² - 11² - 9²

max_d = √454 - 196 - 121 - 81

max_d = √56

max_d = 7.4833147735479

Since max_d = 7.4833147735479 is not an integer, this is not a solution

c = 10

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

max_d = √454 - 14² - 11² - 10²

max_d = √454 - 196 - 121 - 100

max_d = √37

max_d = 6.0827625302982

Since max_d = 6.0827625302982 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 = √454 - 14² - 11² - 11²

max_d = √454 - 196 - 121 - 121

max_d = √16

max_d = 4

Since max_d = 4, then (a, b, c, d) = (14, 11, 11, 4) is an integer solution proven below

14² + 11² + 11² + 4² → 196 + 121 + 121 + 16 = 454

b = 12

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 14² - 12²)

max_c = Floor(√454 - 196 - 144)

max_c = Floor(√114)

max_c = Floor(10.677078252031)

max_c = 10

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 14² - 12²)/2 = 57

When min_c = 8, then it is c² = 64 ≥ 57, so min_c = 8

c = 8

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

max_d = √454 - 14² - 12² - 8²

max_d = √454 - 196 - 144 - 64

max_d = √50

max_d = 7.0710678118655

Since max_d = 7.0710678118655 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 = √454 - 14² - 12² - 9²

max_d = √454 - 196 - 144 - 81

max_d = √33

max_d = 5.744562646538

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

c = 10

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

max_d = √454 - 14² - 12² - 10²

max_d = √454 - 196 - 144 - 100

max_d = √14

max_d = 3.7416573867739

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

b = 13

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 14² - 13²)

max_c = Floor(√454 - 196 - 169)

max_c = Floor(√89)

max_c = Floor(9.4339811320566)

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 = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 14² - 13²)/2 = 44.5

When min_c = 7, then it is c² = 49 ≥ 44.5, so min_c = 7

c = 7

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

max_d = √454 - 14² - 13² - 7²

max_d = √454 - 196 - 169 - 49

max_d = √40

max_d = 6.3245553203368

Since max_d = 6.3245553203368 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 = √454 - 14² - 13² - 8²

max_d = √454 - 196 - 169 - 64

max_d = √25

max_d = 5

Since max_d = 5, then (a, b, c, d) = (14, 13, 8, 5) is an integer solution proven below

14² + 13² + 8² + 5² → 196 + 169 + 64 + 25 = 454

c = 9

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

max_d = √454 - 14² - 13² - 9²

max_d = √454 - 196 - 169 - 81

max_d = √8

max_d = 2.8284271247462

Since max_d = 2.8284271247462 is not an integer, this is not a solution

b = 14

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 14² - 14²)

max_c = Floor(√454 - 196 - 196)

max_c = Floor(√62)

max_c = Floor(7.8740078740118)

max_c = 7

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 14² - 14²)/2 = 31

When min_c = 6, then it is c² = 36 ≥ 31, so min_c = 6

c = 6

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

max_d = √454 - 14² - 14² - 6²

max_d = √454 - 196 - 196 - 36

max_d = √26

max_d = 5.0990195135928

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

c = 7

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

max_d = √454 - 14² - 14² - 7²

max_d = √454 - 196 - 196 - 49

max_d = √13

max_d = 3.605551275464

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

b = 15

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 14² - 15²)

max_c = Floor(√454 - 196 - 225)

max_c = Floor(√33)

max_c = Floor(5.744562646538)

max_c = 5

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 14² - 15²)/2 = 16.5

When min_c = 5, then it is c² = 25 ≥ 16.5, so min_c = 5

c = 5

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

max_d = √454 - 14² - 15² - 5²

max_d = √454 - 196 - 225 - 25

max_d = √8

max_d = 2.8284271247462

Since max_d = 2.8284271247462 is not an integer, this is not a solution

b = 16

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 14² - 16²)

max_c = Floor(√454 - 196 - 256)

max_c = Floor(√2)

max_c = Floor(1.4142135623731)

max_c = 1

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 14² - 16²)/2 = 1

When min_c = 1, then it is c² = 1 ≥ 1, so min_c = 1

c = 1

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

max_d = √454 - 14² - 16² - 1²

max_d = √454 - 196 - 256 - 1

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (14, 16, 1, 1) is an integer solution proven below

14² + 16² + 1² + 1² → 196 + 256 + 1 + 1 = 454

a = 15

Find max_b which is Floor(√n - a²)

max_b = Floor(√454 - 15²)

max_b = Floor(√454 - 225)

max_b = Floor(√229)

max_b = Floor(15.132745950422)

max_b = 15

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 1 and increase by 1
Go until (n - a²)/3 → (454 - 15²)/3 = 76.333333333333

When min_b = 9, then it is b² = 81 ≥ 76.333333333333, so min_b = 9

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

(9, 15)

b = 9

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 15² - 9²)

max_c = Floor(√454 - 225 - 81)

max_c = Floor(√148)

max_c = Floor(12.165525060596)

max_c = 12

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 15² - 9²)/2 = 74

When min_c = 9, then it is c² = 81 ≥ 74, so min_c = 9

c = 9

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

max_d = √454 - 15² - 9² - 9²

max_d = √454 - 225 - 81 - 81

max_d = √67

max_d = 8.1853527718725

Since max_d = 8.1853527718725 is not an integer, this is not a solution

c = 10

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

max_d = √454 - 15² - 9² - 10²

max_d = √454 - 225 - 81 - 100

max_d = √48

max_d = 6.9282032302755

Since max_d = 6.9282032302755 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 = √454 - 15² - 9² - 11²

max_d = √454 - 225 - 81 - 121

max_d = √27

max_d = 5.1961524227066

Since max_d = 5.1961524227066 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 = √454 - 15² - 9² - 12²

max_d = √454 - 225 - 81 - 144

max_d = √4

max_d = 2

Since max_d = 2, then (a, b, c, d) = (15, 9, 12, 2) is an integer solution proven below

15² + 9² + 12² + 2² → 225 + 81 + 144 + 4 = 454

b = 10

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 15² - 10²)

max_c = Floor(√454 - 225 - 100)

max_c = Floor(√129)

max_c = Floor(11.357816691601)

max_c = 11

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 15² - 10²)/2 = 64.5

When min_c = 9, then it is c² = 81 ≥ 64.5, so min_c = 9

c = 9

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

max_d = √454 - 15² - 10² - 9²

max_d = √454 - 225 - 100 - 81

max_d = √48

max_d = 6.9282032302755

Since max_d = 6.9282032302755 is not an integer, this is not a solution

c = 10

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

max_d = √454 - 15² - 10² - 10²

max_d = √454 - 225 - 100 - 100

max_d = √29

max_d = 5.3851648071345

Since max_d = 5.3851648071345 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 = √454 - 15² - 10² - 11²

max_d = √454 - 225 - 100 - 121

max_d = √8

max_d = 2.8284271247462

Since max_d = 2.8284271247462 is not an integer, this is not a solution

b = 11

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 15² - 11²)

max_c = Floor(√454 - 225 - 121)

max_c = Floor(√108)

max_c = Floor(10.392304845413)

max_c = 10

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 15² - 11²)/2 = 54

When min_c = 8, then it is c² = 64 ≥ 54, so min_c = 8

c = 8

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

max_d = √454 - 15² - 11² - 8²

max_d = √454 - 225 - 121 - 64

max_d = √44

max_d = 6.6332495807108

Since max_d = 6.6332495807108 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 = √454 - 15² - 11² - 9²

max_d = √454 - 225 - 121 - 81

max_d = √27

max_d = 5.1961524227066

Since max_d = 5.1961524227066 is not an integer, this is not a solution

c = 10

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

max_d = √454 - 15² - 11² - 10²

max_d = √454 - 225 - 121 - 100

max_d = √8

max_d = 2.8284271247462

Since max_d = 2.8284271247462 is not an integer, this is not a solution

b = 12

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 15² - 12²)

max_c = Floor(√454 - 225 - 144)

max_c = Floor(√85)

max_c = Floor(9.2195444572929)

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 = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 15² - 12²)/2 = 42.5

When min_c = 7, then it is c² = 49 ≥ 42.5, so min_c = 7

c = 7

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

max_d = √454 - 15² - 12² - 7²

max_d = √454 - 225 - 144 - 49

max_d = √36

max_d = 6

Since max_d = 6, then (a, b, c, d) = (15, 12, 7, 6) is an integer solution proven below

15² + 12² + 7² + 6² → 225 + 144 + 49 + 36 = 454

c = 8

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

max_d = √454 - 15² - 12² - 8²

max_d = √454 - 225 - 144 - 64

max_d = √21

max_d = 4.5825756949558

Since max_d = 4.5825756949558 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 = √454 - 15² - 12² - 9²

max_d = √454 - 225 - 144 - 81

max_d = √4

max_d = 2

Since max_d = 2, then (a, b, c, d) = (15, 12, 9, 2) is an integer solution proven below

15² + 12² + 9² + 2² → 225 + 144 + 81 + 4 = 454

b = 13

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 15² - 13²)

max_c = Floor(√454 - 225 - 169)

max_c = Floor(√60)

max_c = Floor(7.7459666924148)

max_c = 7

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 15² - 13²)/2 = 30

When min_c = 6, then it is c² = 36 ≥ 30, so min_c = 6

c = 6

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

max_d = √454 - 15² - 13² - 6²

max_d = √454 - 225 - 169 - 36

max_d = √24

max_d = 4.8989794855664

Since max_d = 4.8989794855664 is not an integer, this is not a solution

c = 7

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

max_d = √454 - 15² - 13² - 7²

max_d = √454 - 225 - 169 - 49

max_d = √11

max_d = 3.3166247903554

Since max_d = 3.3166247903554 is not an integer, this is not a solution

b = 14

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 15² - 14²)

max_c = Floor(√454 - 225 - 196)

max_c = Floor(√33)

max_c = Floor(5.744562646538)

max_c = 5

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 15² - 14²)/2 = 16.5

When min_c = 5, then it is c² = 25 ≥ 16.5, so min_c = 5

c = 5

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

max_d = √454 - 15² - 14² - 5²

max_d = √454 - 225 - 196 - 25

max_d = √8

max_d = 2.8284271247462

Since max_d = 2.8284271247462 is not an integer, this is not a solution

b = 15

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 15² - 15²)

max_c = Floor(√454 - 225 - 225)

max_c = Floor(√4)

max_c = Floor(2)

max_c = 2

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 15² - 15²)/2 = 2

When min_c = 2, then it is c² = 4 ≥ 2, so min_c = 2

c = 2

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

max_d = √454 - 15² - 15² - 2²

max_d = √454 - 225 - 225 - 4

max_d = √0

max_d = 0

Since max_d = 0, then (a, b, c, d) = (15, 15, 2, 0) is an integer solution proven below

15² + 15² + 2² + 0² → 225 + 225 + 4 + 0 = 454

a = 16

Find max_b which is Floor(√n - a²)

max_b = Floor(√454 - 16²)

max_b = Floor(√454 - 256)

max_b = Floor(√198)

max_b = Floor(14.07124727947)

max_b = 14

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 1 and increase by 1
Go until (n - a²)/3 → (454 - 16²)/3 = 66

When min_b = 9, then it is b² = 81 ≥ 66, so min_b = 9

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

(9, 14)

b = 9

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 16² - 9²)

max_c = Floor(√454 - 256 - 81)

max_c = Floor(√117)

max_c = Floor(10.816653826392)

max_c = 10

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 16² - 9²)/2 = 58.5

When min_c = 8, then it is c² = 64 ≥ 58.5, so min_c = 8

c = 8

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

max_d = √454 - 16² - 9² - 8²

max_d = √454 - 256 - 81 - 64

max_d = √53

max_d = 7.2801098892805

Since max_d = 7.2801098892805 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 = √454 - 16² - 9² - 9²

max_d = √454 - 256 - 81 - 81

max_d = √36

max_d = 6

Since max_d = 6, then (a, b, c, d) = (16, 9, 9, 6) is an integer solution proven below

16² + 9² + 9² + 6² → 256 + 81 + 81 + 36 = 454

c = 10

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

max_d = √454 - 16² - 9² - 10²

max_d = √454 - 256 - 81 - 100

max_d = √17

max_d = 4.1231056256177

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

b = 10

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 16² - 10²)

max_c = Floor(√454 - 256 - 100)

max_c = Floor(√98)

max_c = Floor(9.8994949366117)

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 = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 16² - 10²)/2 = 49

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

c = 7

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

max_d = √454 - 16² - 10² - 7²

max_d = √454 - 256 - 100 - 49

max_d = √49

max_d = 7

Since max_d = 7, then (a, b, c, d) = (16, 10, 7, 7) is an integer solution proven below

16² + 10² + 7² + 7² → 256 + 100 + 49 + 49 = 454

c = 8

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

max_d = √454 - 16² - 10² - 8²

max_d = √454 - 256 - 100 - 64

max_d = √34

max_d = 5.8309518948453

Since max_d = 5.8309518948453 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 = √454 - 16² - 10² - 9²

max_d = √454 - 256 - 100 - 81

max_d = √17

max_d = 4.1231056256177

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

b = 11

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 16² - 11²)

max_c = Floor(√454 - 256 - 121)

max_c = Floor(√77)

max_c = Floor(8.7749643873921)

max_c = 8

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 16² - 11²)/2 = 38.5

When min_c = 7, then it is c² = 49 ≥ 38.5, so min_c = 7

c = 7

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

max_d = √454 - 16² - 11² - 7²

max_d = √454 - 256 - 121 - 49

max_d = √28

max_d = 5.2915026221292

Since max_d = 5.2915026221292 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 = √454 - 16² - 11² - 8²

max_d = √454 - 256 - 121 - 64

max_d = √13

max_d = 3.605551275464

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

b = 12

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 16² - 12²)

max_c = Floor(√454 - 256 - 144)

max_c = Floor(√54)

max_c = Floor(7.3484692283495)

max_c = 7

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 16² - 12²)/2 = 27

When min_c = 6, then it is c² = 36 ≥ 27, so min_c = 6

c = 6

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

max_d = √454 - 16² - 12² - 6²

max_d = √454 - 256 - 144 - 36

max_d = √18

max_d = 4.2426406871193

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

c = 7

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

max_d = √454 - 16² - 12² - 7²

max_d = √454 - 256 - 144 - 49

max_d = √5

max_d = 2.2360679774998

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

b = 13

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 16² - 13²)

max_c = Floor(√454 - 256 - 169)

max_c = Floor(√29)

max_c = Floor(5.3851648071345)

max_c = 5

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 16² - 13²)/2 = 14.5

When min_c = 4, then it is c² = 16 ≥ 14.5, so min_c = 4

c = 4

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

max_d = √454 - 16² - 13² - 4²

max_d = √454 - 256 - 169 - 16

max_d = √13

max_d = 3.605551275464

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

c = 5

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

max_d = √454 - 16² - 13² - 5²

max_d = √454 - 256 - 169 - 25

max_d = √4

max_d = 2

Since max_d = 2, then (a, b, c, d) = (16, 13, 5, 2) is an integer solution proven below

16² + 13² + 5² + 2² → 256 + 169 + 25 + 4 = 454

b = 14

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 16² - 14²)

max_c = Floor(√454 - 256 - 196)

max_c = Floor(√2)

max_c = Floor(1.4142135623731)

max_c = 1

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 16² - 14²)/2 = 1

When min_c = 1, then it is c² = 1 ≥ 1, so min_c = 1

c = 1

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

max_d = √454 - 16² - 14² - 1²

max_d = √454 - 256 - 196 - 1

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (16, 14, 1, 1) is an integer solution proven below

16² + 14² + 1² + 1² → 256 + 196 + 1 + 1 = 454

a = 17

Find max_b which is Floor(√n - a²)

max_b = Floor(√454 - 17²)

max_b = Floor(√454 - 289)

max_b = Floor(√165)

max_b = Floor(12.845232578665)

max_b = 12

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 1 and increase by 1
Go until (n - a²)/3 → (454 - 17²)/3 = 55

When min_b = 8, then it is b² = 64 ≥ 55, so min_b = 8

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

(8, 12)

b = 8

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 17² - 8²)

max_c = Floor(√454 - 289 - 64)

max_c = Floor(√101)

max_c = Floor(10.049875621121)

max_c = 10

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 17² - 8²)/2 = 50.5

When min_c = 8, then it is c² = 64 ≥ 50.5, so min_c = 8

c = 8

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

max_d = √454 - 17² - 8² - 8²

max_d = √454 - 289 - 64 - 64

max_d = √37

max_d = 6.0827625302982

Since max_d = 6.0827625302982 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 = √454 - 17² - 8² - 9²

max_d = √454 - 289 - 64 - 81

max_d = √20

max_d = 4.4721359549996

Since max_d = 4.4721359549996 is not an integer, this is not a solution

c = 10

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

max_d = √454 - 17² - 8² - 10²

max_d = √454 - 289 - 64 - 100

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (17, 8, 10, 1) is an integer solution proven below

17² + 8² + 10² + 1² → 289 + 64 + 100 + 1 = 454

b = 9

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 17² - 9²)

max_c = Floor(√454 - 289 - 81)

max_c = Floor(√84)

max_c = Floor(9.1651513899117)

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 = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 17² - 9²)/2 = 42

When min_c = 7, then it is c² = 49 ≥ 42, so min_c = 7

c = 7

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

max_d = √454 - 17² - 9² - 7²

max_d = √454 - 289 - 81 - 49

max_d = √35

max_d = 5.9160797830996

Since max_d = 5.9160797830996 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 = √454 - 17² - 9² - 8²

max_d = √454 - 289 - 81 - 64

max_d = √20

max_d = 4.4721359549996

Since max_d = 4.4721359549996 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 = √454 - 17² - 9² - 9²

max_d = √454 - 289 - 81 - 81

max_d = √3

max_d = 1.7320508075689

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

b = 10

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 17² - 10²)

max_c = Floor(√454 - 289 - 100)

max_c = Floor(√65)

max_c = Floor(8.0622577482985)

max_c = 8

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 17² - 10²)/2 = 32.5

When min_c = 6, then it is c² = 36 ≥ 32.5, so min_c = 6

c = 6

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

max_d = √454 - 17² - 10² - 6²

max_d = √454 - 289 - 100 - 36

max_d = √29

max_d = 5.3851648071345

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

c = 7

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

max_d = √454 - 17² - 10² - 7²

max_d = √454 - 289 - 100 - 49

max_d = √16

max_d = 4

Since max_d = 4, then (a, b, c, d) = (17, 10, 7, 4) is an integer solution proven below

17² + 10² + 7² + 4² → 289 + 100 + 49 + 16 = 454

c = 8

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

max_d = √454 - 17² - 10² - 8²

max_d = √454 - 289 - 100 - 64

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (17, 10, 8, 1) is an integer solution proven below

17² + 10² + 8² + 1² → 289 + 100 + 64 + 1 = 454

b = 11

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 17² - 11²)

max_c = Floor(√454 - 289 - 121)

max_c = Floor(√44)

max_c = Floor(6.6332495807108)

max_c = 6

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 17² - 11²)/2 = 22

When min_c = 5, then it is c² = 25 ≥ 22, so min_c = 5

c = 5

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

max_d = √454 - 17² - 11² - 5²

max_d = √454 - 289 - 121 - 25

max_d = √19

max_d = 4.3588989435407

Since max_d = 4.3588989435407 is not an integer, this is not a solution

c = 6

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

max_d = √454 - 17² - 11² - 6²

max_d = √454 - 289 - 121 - 36

max_d = √8

max_d = 2.8284271247462

Since max_d = 2.8284271247462 is not an integer, this is not a solution

b = 12

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 17² - 12²)

max_c = Floor(√454 - 289 - 144)

max_c = Floor(√21)

max_c = Floor(4.5825756949558)

max_c = 4

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 17² - 12²)/2 = 10.5

When min_c = 4, then it is c² = 16 ≥ 10.5, so min_c = 4

c = 4

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

max_d = √454 - 17² - 12² - 4²

max_d = √454 - 289 - 144 - 16

max_d = √5

max_d = 2.2360679774998

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

a = 18

Find max_b which is Floor(√n - a²)

max_b = Floor(√454 - 18²)

max_b = Floor(√454 - 324)

max_b = Floor(√130)

max_b = Floor(11.401754250991)

max_b = 11

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 1 and increase by 1
Go until (n - a²)/3 → (454 - 18²)/3 = 43.333333333333

When min_b = 7, then it is b² = 49 ≥ 43.333333333333, so min_b = 7

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

(7, 11)

b = 7

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 18² - 7²)

max_c = Floor(√454 - 324 - 49)

max_c = Floor(√81)

max_c = Floor(9)

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 = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 18² - 7²)/2 = 40.5

When min_c = 7, then it is c² = 49 ≥ 40.5, so min_c = 7

c = 7

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

max_d = √454 - 18² - 7² - 7²

max_d = √454 - 324 - 49 - 49

max_d = √32

max_d = 5.6568542494924

Since max_d = 5.6568542494924 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 = √454 - 18² - 7² - 8²

max_d = √454 - 324 - 49 - 64

max_d = √17

max_d = 4.1231056256177

Since max_d = 4.1231056256177 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 = √454 - 18² - 7² - 9²

max_d = √454 - 324 - 49 - 81

max_d = √0

max_d = 0

Since max_d = 0, then (a, b, c, d) = (18, 7, 9, 0) is an integer solution proven below

18² + 7² + 9² + 0² → 324 + 49 + 81 + 0 = 454

b = 8

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 18² - 8²)

max_c = Floor(√454 - 324 - 64)

max_c = Floor(√66)

max_c = Floor(8.124038404636)

max_c = 8

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 18² - 8²)/2 = 33

When min_c = 6, then it is c² = 36 ≥ 33, so min_c = 6

c = 6

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

max_d = √454 - 18² - 8² - 6²

max_d = √454 - 324 - 64 - 36

max_d = √30

max_d = 5.4772255750517

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

c = 7

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

max_d = √454 - 18² - 8² - 7²

max_d = √454 - 324 - 64 - 49

max_d = √17

max_d = 4.1231056256177

Since max_d = 4.1231056256177 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 = √454 - 18² - 8² - 8²

max_d = √454 - 324 - 64 - 64

max_d = √2

max_d = 1.4142135623731

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

b = 9

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 18² - 9²)

max_c = Floor(√454 - 324 - 81)

max_c = Floor(√49)

max_c = Floor(7)

max_c = 7

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 18² - 9²)/2 = 24.5

When min_c = 5, then it is c² = 25 ≥ 24.5, so min_c = 5

c = 5

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

max_d = √454 - 18² - 9² - 5²

max_d = √454 - 324 - 81 - 25

max_d = √24

max_d = 4.8989794855664

Since max_d = 4.8989794855664 is not an integer, this is not a solution

c = 6

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

max_d = √454 - 18² - 9² - 6²

max_d = √454 - 324 - 81 - 36

max_d = √13

max_d = 3.605551275464

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

c = 7

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

max_d = √454 - 18² - 9² - 7²

max_d = √454 - 324 - 81 - 49

max_d = √0

max_d = 0

Since max_d = 0, then (a, b, c, d) = (18, 9, 7, 0) is an integer solution proven below

18² + 9² + 7² + 0² → 324 + 81 + 49 + 0 = 454

b = 10

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 18² - 10²)

max_c = Floor(√454 - 324 - 100)

max_c = Floor(√30)

max_c = Floor(5.4772255750517)

max_c = 5

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 18² - 10²)/2 = 15

When min_c = 4, then it is c² = 16 ≥ 15, so min_c = 4

c = 4

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

max_d = √454 - 18² - 10² - 4²

max_d = √454 - 324 - 100 - 16

max_d = √14

max_d = 3.7416573867739

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

c = 5

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

max_d = √454 - 18² - 10² - 5²

max_d = √454 - 324 - 100 - 25

max_d = √5

max_d = 2.2360679774998

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

b = 11

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 18² - 11²)

max_c = Floor(√454 - 324 - 121)

max_c = Floor(√9)

max_c = Floor(3)

max_c = 3

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 18² - 11²)/2 = 4.5

When min_c = 3, then it is c² = 9 ≥ 4.5, so min_c = 3

c = 3

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

max_d = √454 - 18² - 11² - 3²

max_d = √454 - 324 - 121 - 9

max_d = √0

max_d = 0

Since max_d = 0, then (a, b, c, d) = (18, 11, 3, 0) is an integer solution proven below

18² + 11² + 3² + 0² → 324 + 121 + 9 + 0 = 454

a = 19

Find max_b which is Floor(√n - a²)

max_b = Floor(√454 - 19²)

max_b = Floor(√454 - 361)

max_b = Floor(√93)

max_b = Floor(9.643650760993)

max_b = 9

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 1 and increase by 1
Go until (n - a²)/3 → (454 - 19²)/3 = 31

When min_b = 6, then it is b² = 36 ≥ 31, so min_b = 6

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

(6, 9)

b = 6

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 19² - 6²)

max_c = Floor(√454 - 361 - 36)

max_c = Floor(√57)

max_c = Floor(7.5498344352707)

max_c = 7

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 19² - 6²)/2 = 28.5

When min_c = 6, then it is c² = 36 ≥ 28.5, so min_c = 6

c = 6

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

max_d = √454 - 19² - 6² - 6²

max_d = √454 - 361 - 36 - 36

max_d = √21

max_d = 4.5825756949558

Since max_d = 4.5825756949558 is not an integer, this is not a solution

c = 7

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

max_d = √454 - 19² - 6² - 7²

max_d = √454 - 361 - 36 - 49

max_d = √8

max_d = 2.8284271247462

Since max_d = 2.8284271247462 is not an integer, this is not a solution

b = 7

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 19² - 7²)

max_c = Floor(√454 - 361 - 49)

max_c = Floor(√44)

max_c = Floor(6.6332495807108)

max_c = 6

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 19² - 7²)/2 = 22

When min_c = 5, then it is c² = 25 ≥ 22, so min_c = 5

c = 5

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

max_d = √454 - 19² - 7² - 5²

max_d = √454 - 361 - 49 - 25

max_d = √19

max_d = 4.3588989435407

Since max_d = 4.3588989435407 is not an integer, this is not a solution

c = 6

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

max_d = √454 - 19² - 7² - 6²

max_d = √454 - 361 - 49 - 36

max_d = √8

max_d = 2.8284271247462

Since max_d = 2.8284271247462 is not an integer, this is not a solution

b = 8

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 19² - 8²)

max_c = Floor(√454 - 361 - 64)

max_c = Floor(√29)

max_c = Floor(5.3851648071345)

max_c = 5

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 19² - 8²)/2 = 14.5

When min_c = 4, then it is c² = 16 ≥ 14.5, so min_c = 4

c = 4

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

max_d = √454 - 19² - 8² - 4²

max_d = √454 - 361 - 64 - 16

max_d = √13

max_d = 3.605551275464

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

c = 5

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

max_d = √454 - 19² - 8² - 5²

max_d = √454 - 361 - 64 - 25

max_d = √4

max_d = 2

Since max_d = 2, then (a, b, c, d) = (19, 8, 5, 2) is an integer solution proven below

19² + 8² + 5² + 2² → 361 + 64 + 25 + 4 = 454

b = 9

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 19² - 9²)

max_c = Floor(√454 - 361 - 81)

max_c = Floor(√12)

max_c = Floor(3.4641016151378)

max_c = 3

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 19² - 9²)/2 = 6

When min_c = 3, then it is c² = 9 ≥ 6, so min_c = 3

c = 3

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

max_d = √454 - 19² - 9² - 3²

max_d = √454 - 361 - 81 - 9

max_d = √3

max_d = 1.7320508075689

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

a = 20

Find max_b which is Floor(√n - a²)

max_b = Floor(√454 - 20²)

max_b = Floor(√454 - 400)

max_b = Floor(√54)

max_b = Floor(7.3484692283495)

max_b = 7

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 1 and increase by 1
Go until (n - a²)/3 → (454 - 20²)/3 = 18

When min_b = 5, then it is b² = 25 ≥ 18, so min_b = 5

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

(5, 7)

b = 5

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 20² - 5²)

max_c = Floor(√454 - 400 - 25)

max_c = Floor(√29)

max_c = Floor(5.3851648071345)

max_c = 5

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 20² - 5²)/2 = 14.5

When min_c = 4, then it is c² = 16 ≥ 14.5, so min_c = 4

c = 4

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

max_d = √454 - 20² - 5² - 4²

max_d = √454 - 400 - 25 - 16

max_d = √13

max_d = 3.605551275464

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

c = 5

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

max_d = √454 - 20² - 5² - 5²

max_d = √454 - 400 - 25 - 25

max_d = √4

max_d = 2

Since max_d = 2, then (a, b, c, d) = (20, 5, 5, 2) is an integer solution proven below

20² + 5² + 5² + 2² → 400 + 25 + 25 + 4 = 454

b = 6

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 20² - 6²)

max_c = Floor(√454 - 400 - 36)

max_c = Floor(√18)

max_c = Floor(4.2426406871193)

max_c = 4

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 20² - 6²)/2 = 9

When min_c = 3, then it is c² = 9 ≥ 9, so min_c = 3

c = 3

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

max_d = √454 - 20² - 6² - 3²

max_d = √454 - 400 - 36 - 9

max_d = √9

max_d = 3

Since max_d = 3, then (a, b, c, d) = (20, 6, 3, 3) is an integer solution proven below

20² + 6² + 3² + 3² → 400 + 36 + 9 + 9 = 454

c = 4

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

max_d = √454 - 20² - 6² - 4²

max_d = √454 - 400 - 36 - 16

max_d = √2

max_d = 1.4142135623731

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

b = 7

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 20² - 7²)

max_c = Floor(√454 - 400 - 49)

max_c = Floor(√5)

max_c = Floor(2.2360679774998)

max_c = 2

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 20² - 7²)/2 = 2.5

When min_c = 2, then it is c² = 4 ≥ 2.5, so min_c = 2

c = 2

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

max_d = √454 - 20² - 7² - 2²

max_d = √454 - 400 - 49 - 4

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (20, 7, 2, 1) is an integer solution proven below

20² + 7² + 2² + 1² → 400 + 49 + 4 + 1 = 454

a = 21

Find max_b which is Floor(√n - a²)

max_b = Floor(√454 - 21²)

max_b = Floor(√454 - 441)

max_b = Floor(√13)

max_b = Floor(3.605551275464)

max_b = 3

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 1 and increase by 1
Go until (n - a²)/3 → (454 - 21²)/3 = 4.3333333333333

When min_b = 3, then it is b² = 9 ≥ 4.3333333333333, so min_b = 3

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

(3, 3)

b = 3

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√454 - 21² - 3²)

max_c = Floor(√454 - 441 - 9)

max_c = Floor(√4)

max_c = Floor(2)

max_c = 2

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

Call it min_b

Start with min_c = 1 and increase by 1

Go until (n - a² - b² )/2 → (454 - 21² - 3²)/2 = 2

When min_c = 2, then it is c² = 4 ≥ 2, so min_c = 2

c = 2

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

max_d = √454 - 21² - 3² - 2²

max_d = √454 - 441 - 9 - 4

max_d = √0

max_d = 0

Since max_d = 0, then (a, b, c, d) = (21, 3, 2, 0) is an integer solution proven below

21² + 3² + 2² + 0² → 441 + 9 + 4 + 0 = 454

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Tags:

• algorithm
• floor
• integer
• lagrange theorem
• maximum
• minimum
• natural number