Lagrange four-square (Bachet Conjecture) for 300

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Show the Lagrange Four Square Theorem for

300

Lagrange Four Square Definition

For any natural number (p), we write as

p = a² + b² + c² + d²

Determine max_a:

Floor(√300) = Floor(17.320508075689)

Floor(17.320508075689) = 17
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 → 300/4 = 75

When min_a = 9, then it is a² = 81 ≥ 75, so min_a = 9

Find a in the range of (min_a, max_a)

(9, 17)

a = 9

Find max_b which is Floor(√n - a²)

max_b = Floor(√300 - 9²)

max_b = Floor(√300 - 81)

max_b = Floor(√219)

max_b = Floor(14.798648586949)

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 → (300 - 9²)/3 = 73

When min_b = 9, then it is b² = 81 ≥ 73, 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(√300 - 9² - 9²)

max_c = Floor(√300 - 81 - 81)

max_c = Floor(√138)

max_c = Floor(11.747340124471)

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 → (300 - 9² - 9²)/2 = 69

When min_c = 9, then it is c² = 81 ≥ 69, so min_c = 9

c = 9

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

max_d = √300 - 9² - 9² - 9²

max_d = √300 - 81 - 81 - 81

max_d = √57

max_d = 7.5498344352707

Since max_d = 7.5498344352707 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 = √300 - 9² - 9² - 10²

max_d = √300 - 81 - 81 - 100

max_d = √38

max_d = 6.164414002969

Since max_d = 6.164414002969 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 = √300 - 9² - 9² - 11²

max_d = √300 - 81 - 81 - 121

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(√300 - 9² - 10²)

max_c = Floor(√300 - 81 - 100)

max_c = Floor(√119)

max_c = Floor(10.908712114636)

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 → (300 - 9² - 10²)/2 = 59.5

When min_c = 8, then it is c² = 64 ≥ 59.5, so min_c = 8

c = 8

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

max_d = √300 - 9² - 10² - 8²

max_d = √300 - 81 - 100 - 64

max_d = √55

max_d = 7.4161984870957

Since max_d = 7.4161984870957 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 = √300 - 9² - 10² - 9²

max_d = √300 - 81 - 100 - 81

max_d = √38

max_d = 6.164414002969

Since max_d = 6.164414002969 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 = √300 - 9² - 10² - 10²

max_d = √300 - 81 - 100 - 100

max_d = √19

max_d = 4.3588989435407

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

b = 11

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 9² - 11²)

max_c = Floor(√300 - 81 - 121)

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 → (300 - 9² - 11²)/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 = √300 - 9² - 11² - 7²

max_d = √300 - 81 - 121 - 49

max_d = √49

max_d = 7

Since max_d = 7, then (a, b, c, d) = (9, 11, 7, 7) is an integer solution proven below

9² + 11² + 7² + 7² → 81 + 121 + 49 + 49 = 300

c = 8

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

max_d = √300 - 9² - 11² - 8²

max_d = √300 - 81 - 121 - 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 = √300 - 9² - 11² - 9²

max_d = √300 - 81 - 121 - 81

max_d = √17

max_d = 4.1231056256177

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

b = 12

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 9² - 12²)

max_c = Floor(√300 - 81 - 144)

max_c = Floor(√75)

max_c = Floor(8.6602540378444)

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 → (300 - 9² - 12²)/2 = 37.5

When min_c = 7, then it is c² = 49 ≥ 37.5, so min_c = 7

c = 7

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

max_d = √300 - 9² - 12² - 7²

max_d = √300 - 81 - 144 - 49

max_d = √26

max_d = 5.0990195135928

Since max_d = 5.0990195135928 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 = √300 - 9² - 12² - 8²

max_d = √300 - 81 - 144 - 64

max_d = √11

max_d = 3.3166247903554

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

b = 13

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 9² - 13²)

max_c = Floor(√300 - 81 - 169)

max_c = Floor(√50)

max_c = Floor(7.0710678118655)

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 → (300 - 9² - 13²)/2 = 25

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

c = 5

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

max_d = √300 - 9² - 13² - 5²

max_d = √300 - 81 - 169 - 25

max_d = √25

max_d = 5

Since max_d = 5, then (a, b, c, d) = (9, 13, 5, 5) is an integer solution proven below

9² + 13² + 5² + 5² → 81 + 169 + 25 + 25 = 300

c = 6

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

max_d = √300 - 9² - 13² - 6²

max_d = √300 - 81 - 169 - 36

max_d = √14

max_d = 3.7416573867739

Since max_d = 3.7416573867739 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 = √300 - 9² - 13² - 7²

max_d = √300 - 81 - 169 - 49

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (9, 13, 7, 1) is an integer solution proven below

9² + 13² + 7² + 1² → 81 + 169 + 49 + 1 = 300

b = 14

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 9² - 14²)

max_c = Floor(√300 - 81 - 196)

max_c = Floor(√23)

max_c = Floor(4.7958315233127)

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 → (300 - 9² - 14²)/2 = 11.5

When min_c = 4, then it is c² = 16 ≥ 11.5, so min_c = 4

c = 4

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

max_d = √300 - 9² - 14² - 4²

max_d = √300 - 81 - 196 - 16

max_d = √7

max_d = 2.6457513110646

Since max_d = 2.6457513110646 is not an integer, this is not a solution

a = 10

Find max_b which is Floor(√n - a²)

max_b = Floor(√300 - 10²)

max_b = Floor(√300 - 100)

max_b = Floor(√200)

max_b = Floor(14.142135623731)

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 → (300 - 10²)/3 = 66.666666666667

When min_b = 9, then it is b² = 81 ≥ 66.666666666667, 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(√300 - 10² - 9²)

max_c = Floor(√300 - 100 - 81)

max_c = Floor(√119)

max_c = Floor(10.908712114636)

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 → (300 - 10² - 9²)/2 = 59.5

When min_c = 8, then it is c² = 64 ≥ 59.5, so min_c = 8

c = 8

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

max_d = √300 - 10² - 9² - 8²

max_d = √300 - 100 - 81 - 64

max_d = √55

max_d = 7.4161984870957

Since max_d = 7.4161984870957 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 = √300 - 10² - 9² - 9²

max_d = √300 - 100 - 81 - 81

max_d = √38

max_d = 6.164414002969

Since max_d = 6.164414002969 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 = √300 - 10² - 9² - 10²

max_d = √300 - 100 - 81 - 100

max_d = √19

max_d = 4.3588989435407

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

b = 10

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 10² - 10²)

max_c = Floor(√300 - 100 - 100)

max_c = Floor(√100)

max_c = Floor(10)

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 → (300 - 10² - 10²)/2 = 50

When min_c = 8, then it is c² = 64 ≥ 50, so min_c = 8

c = 8

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

max_d = √300 - 10² - 10² - 8²

max_d = √300 - 100 - 100 - 64

max_d = √36

max_d = 6

Since max_d = 6, then (a, b, c, d) = (10, 10, 8, 6) is an integer solution proven below

10² + 10² + 8² + 6² → 100 + 100 + 64 + 36 = 300

c = 9

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

max_d = √300 - 10² - 10² - 9²

max_d = √300 - 100 - 100 - 81

max_d = √19

max_d = 4.3588989435407

Since max_d = 4.3588989435407 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 = √300 - 10² - 10² - 10²

max_d = √300 - 100 - 100 - 100

max_d = √0

max_d = 0

Since max_d = 0, then (a, b, c, d) = (10, 10, 10, 0) is an integer solution proven below

10² + 10² + 10² + 0² → 100 + 100 + 100 + 0 = 300

b = 11

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 10² - 11²)

max_c = Floor(√300 - 100 - 121)

max_c = Floor(√79)

max_c = Floor(8.8881944173156)

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 → (300 - 10² - 11²)/2 = 39.5

When min_c = 7, then it is c² = 49 ≥ 39.5, so min_c = 7

c = 7

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

max_d = √300 - 10² - 11² - 7²

max_d = √300 - 100 - 121 - 49

max_d = √30

max_d = 5.4772255750517

Since max_d = 5.4772255750517 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 = √300 - 10² - 11² - 8²

max_d = √300 - 100 - 121 - 64

max_d = √15

max_d = 3.8729833462074

Since max_d = 3.8729833462074 is not an integer, this is not a solution

b = 12

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 10² - 12²)

max_c = Floor(√300 - 100 - 144)

max_c = Floor(√56)

max_c = Floor(7.4833147735479)

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 → (300 - 10² - 12²)/2 = 28

When min_c = 6, then it is c² = 36 ≥ 28, so min_c = 6

c = 6

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

max_d = √300 - 10² - 12² - 6²

max_d = √300 - 100 - 144 - 36

max_d = √20

max_d = 4.4721359549996

Since max_d = 4.4721359549996 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 = √300 - 10² - 12² - 7²

max_d = √300 - 100 - 144 - 49

max_d = √7

max_d = 2.6457513110646

Since max_d = 2.6457513110646 is not an integer, this is not a solution

b = 13

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 10² - 13²)

max_c = Floor(√300 - 100 - 169)

max_c = Floor(√31)

max_c = Floor(5.56776436283)

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 → (300 - 10² - 13²)/2 = 15.5

When min_c = 4, then it is c² = 16 ≥ 15.5, so min_c = 4

c = 4

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

max_d = √300 - 10² - 13² - 4²

max_d = √300 - 100 - 169 - 16

max_d = √15

max_d = 3.8729833462074

Since max_d = 3.8729833462074 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 = √300 - 10² - 13² - 5²

max_d = √300 - 100 - 169 - 25

max_d = √6

max_d = 2.4494897427832

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

b = 14

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 10² - 14²)

max_c = Floor(√300 - 100 - 196)

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 → (300 - 10² - 14²)/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 = √300 - 10² - 14² - 2²

max_d = √300 - 100 - 196 - 4

max_d = √0

max_d = 0

Since max_d = 0, then (a, b, c, d) = (10, 14, 2, 0) is an integer solution proven below

10² + 14² + 2² + 0² → 100 + 196 + 4 + 0 = 300

a = 11

Find max_b which is Floor(√n - a²)

max_b = Floor(√300 - 11²)

max_b = Floor(√300 - 121)

max_b = Floor(√179)

max_b = Floor(13.37908816026)

max_b = 13

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 → (300 - 11²)/3 = 59.666666666667

When min_b = 8, then it is b² = 64 ≥ 59.666666666667, so min_b = 8

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

(8, 13)

b = 8

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 11² - 8²)

max_c = Floor(√300 - 121 - 64)

max_c = Floor(√115)

max_c = Floor(10.723805294764)

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 → (300 - 11² - 8²)/2 = 57.5

When min_c = 8, then it is c² = 64 ≥ 57.5, so min_c = 8

c = 8

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

max_d = √300 - 11² - 8² - 8²

max_d = √300 - 121 - 64 - 64

max_d = √51

max_d = 7.1414284285429

Since max_d = 7.1414284285429 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 = √300 - 11² - 8² - 9²

max_d = √300 - 121 - 64 - 81

max_d = √34

max_d = 5.8309518948453

Since max_d = 5.8309518948453 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 = √300 - 11² - 8² - 10²

max_d = √300 - 121 - 64 - 100

max_d = √15

max_d = 3.8729833462074

Since max_d = 3.8729833462074 is not an integer, this is not a solution

b = 9

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 11² - 9²)

max_c = Floor(√300 - 121 - 81)

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 → (300 - 11² - 9²)/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 = √300 - 11² - 9² - 7²

max_d = √300 - 121 - 81 - 49

max_d = √49

max_d = 7

Since max_d = 7, then (a, b, c, d) = (11, 9, 7, 7) is an integer solution proven below

11² + 9² + 7² + 7² → 121 + 81 + 49 + 49 = 300

c = 8

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

max_d = √300 - 11² - 9² - 8²

max_d = √300 - 121 - 81 - 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 = √300 - 11² - 9² - 9²

max_d = √300 - 121 - 81 - 81

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(√300 - 11² - 10²)

max_c = Floor(√300 - 121 - 100)

max_c = Floor(√79)

max_c = Floor(8.8881944173156)

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 → (300 - 11² - 10²)/2 = 39.5

When min_c = 7, then it is c² = 49 ≥ 39.5, so min_c = 7

c = 7

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

max_d = √300 - 11² - 10² - 7²

max_d = √300 - 121 - 100 - 49

max_d = √30

max_d = 5.4772255750517

Since max_d = 5.4772255750517 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 = √300 - 11² - 10² - 8²

max_d = √300 - 121 - 100 - 64

max_d = √15

max_d = 3.8729833462074

Since max_d = 3.8729833462074 is not an integer, this is not a solution

b = 11

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 11² - 11²)

max_c = Floor(√300 - 121 - 121)

max_c = Floor(√58)

max_c = Floor(7.6157731058639)

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 → (300 - 11² - 11²)/2 = 29

When min_c = 6, then it is c² = 36 ≥ 29, so min_c = 6

c = 6

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

max_d = √300 - 11² - 11² - 6²

max_d = √300 - 121 - 121 - 36

max_d = √22

max_d = 4.6904157598234

Since max_d = 4.6904157598234 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 = √300 - 11² - 11² - 7²

max_d = √300 - 121 - 121 - 49

max_d = √9

max_d = 3

Since max_d = 3, then (a, b, c, d) = (11, 11, 7, 3) is an integer solution proven below

11² + 11² + 7² + 3² → 121 + 121 + 49 + 9 = 300

b = 12

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 11² - 12²)

max_c = Floor(√300 - 121 - 144)

max_c = Floor(√35)

max_c = Floor(5.9160797830996)

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 → (300 - 11² - 12²)/2 = 17.5

When min_c = 5, then it is c² = 25 ≥ 17.5, so min_c = 5

c = 5

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

max_d = √300 - 11² - 12² - 5²

max_d = √300 - 121 - 144 - 25

max_d = √10

max_d = 3.1622776601684

Since max_d = 3.1622776601684 is not an integer, this is not a solution

b = 13

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 11² - 13²)

max_c = Floor(√300 - 121 - 169)

max_c = Floor(√10)

max_c = Floor(3.1622776601684)

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 → (300 - 11² - 13²)/2 = 5

When min_c = 3, then it is c² = 9 ≥ 5, so min_c = 3

c = 3

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

max_d = √300 - 11² - 13² - 3²

max_d = √300 - 121 - 169 - 9

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (11, 13, 3, 1) is an integer solution proven below

11² + 13² + 3² + 1² → 121 + 169 + 9 + 1 = 300

a = 12

Find max_b which is Floor(√n - a²)

max_b = Floor(√300 - 12²)

max_b = Floor(√300 - 144)

max_b = Floor(√156)

max_b = Floor(12.489995996797)

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 → (300 - 12²)/3 = 52

When min_b = 8, then it is b² = 64 ≥ 52, 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(√300 - 12² - 8²)

max_c = Floor(√300 - 144 - 64)

max_c = Floor(√92)

max_c = Floor(9.5916630466254)

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 → (300 - 12² - 8²)/2 = 46

When min_c = 7, then it is c² = 49 ≥ 46, so min_c = 7

c = 7

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

max_d = √300 - 12² - 8² - 7²

max_d = √300 - 144 - 64 - 49

max_d = √43

max_d = 6.557438524302

Since max_d = 6.557438524302 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 = √300 - 12² - 8² - 8²

max_d = √300 - 144 - 64 - 64

max_d = √28

max_d = 5.2915026221292

Since max_d = 5.2915026221292 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 = √300 - 12² - 8² - 9²

max_d = √300 - 144 - 64 - 81

max_d = √11

max_d = 3.3166247903554

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

b = 9

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 12² - 9²)

max_c = Floor(√300 - 144 - 81)

max_c = Floor(√75)

max_c = Floor(8.6602540378444)

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 → (300 - 12² - 9²)/2 = 37.5

When min_c = 7, then it is c² = 49 ≥ 37.5, so min_c = 7

c = 7

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

max_d = √300 - 12² - 9² - 7²

max_d = √300 - 144 - 81 - 49

max_d = √26

max_d = 5.0990195135928

Since max_d = 5.0990195135928 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 = √300 - 12² - 9² - 8²

max_d = √300 - 144 - 81 - 64

max_d = √11

max_d = 3.3166247903554

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

b = 10

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 12² - 10²)

max_c = Floor(√300 - 144 - 100)

max_c = Floor(√56)

max_c = Floor(7.4833147735479)

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 → (300 - 12² - 10²)/2 = 28

When min_c = 6, then it is c² = 36 ≥ 28, so min_c = 6

c = 6

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

max_d = √300 - 12² - 10² - 6²

max_d = √300 - 144 - 100 - 36

max_d = √20

max_d = 4.4721359549996

Since max_d = 4.4721359549996 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 = √300 - 12² - 10² - 7²

max_d = √300 - 144 - 100 - 49

max_d = √7

max_d = 2.6457513110646

Since max_d = 2.6457513110646 is not an integer, this is not a solution

b = 11

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 12² - 11²)

max_c = Floor(√300 - 144 - 121)

max_c = Floor(√35)

max_c = Floor(5.9160797830996)

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 → (300 - 12² - 11²)/2 = 17.5

When min_c = 5, then it is c² = 25 ≥ 17.5, so min_c = 5

c = 5

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

max_d = √300 - 12² - 11² - 5²

max_d = √300 - 144 - 121 - 25

max_d = √10

max_d = 3.1622776601684

Since max_d = 3.1622776601684 is not an integer, this is not a solution

b = 12

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 12² - 12²)

max_c = Floor(√300 - 144 - 144)

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 → (300 - 12² - 12²)/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 = √300 - 12² - 12² - 3²

max_d = √300 - 144 - 144 - 9

max_d = √3

max_d = 1.7320508075689

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

a = 13

Find max_b which is Floor(√n - a²)

max_b = Floor(√300 - 13²)

max_b = Floor(√300 - 169)

max_b = Floor(√131)

max_b = Floor(11.44552314226)

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 → (300 - 13²)/3 = 43.666666666667

When min_b = 7, then it is b² = 49 ≥ 43.666666666667, 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(√300 - 13² - 7²)

max_c = Floor(√300 - 169 - 49)

max_c = Floor(√82)

max_c = Floor(9.0553851381374)

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 → (300 - 13² - 7²)/2 = 41

When min_c = 7, then it is c² = 49 ≥ 41, so min_c = 7

c = 7

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

max_d = √300 - 13² - 7² - 7²

max_d = √300 - 169 - 49 - 49

max_d = √33

max_d = 5.744562646538

Since max_d = 5.744562646538 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 = √300 - 13² - 7² - 8²

max_d = √300 - 169 - 49 - 64

max_d = √18

max_d = 4.2426406871193

Since max_d = 4.2426406871193 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 = √300 - 13² - 7² - 9²

max_d = √300 - 169 - 49 - 81

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (13, 7, 9, 1) is an integer solution proven below

13² + 7² + 9² + 1² → 169 + 49 + 81 + 1 = 300

b = 8

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 13² - 8²)

max_c = Floor(√300 - 169 - 64)

max_c = Floor(√67)

max_c = Floor(8.1853527718725)

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 → (300 - 13² - 8²)/2 = 33.5

When min_c = 6, then it is c² = 36 ≥ 33.5, so min_c = 6

c = 6

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

max_d = √300 - 13² - 8² - 6²

max_d = √300 - 169 - 64 - 36

max_d = √31

max_d = 5.56776436283

Since max_d = 5.56776436283 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 = √300 - 13² - 8² - 7²

max_d = √300 - 169 - 64 - 49

max_d = √18

max_d = 4.2426406871193

Since max_d = 4.2426406871193 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 = √300 - 13² - 8² - 8²

max_d = √300 - 169 - 64 - 64

max_d = √3

max_d = 1.7320508075689

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

b = 9

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 13² - 9²)

max_c = Floor(√300 - 169 - 81)

max_c = Floor(√50)

max_c = Floor(7.0710678118655)

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 → (300 - 13² - 9²)/2 = 25

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

c = 5

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

max_d = √300 - 13² - 9² - 5²

max_d = √300 - 169 - 81 - 25

max_d = √25

max_d = 5

Since max_d = 5, then (a, b, c, d) = (13, 9, 5, 5) is an integer solution proven below

13² + 9² + 5² + 5² → 169 + 81 + 25 + 25 = 300

c = 6

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

max_d = √300 - 13² - 9² - 6²

max_d = √300 - 169 - 81 - 36

max_d = √14

max_d = 3.7416573867739

Since max_d = 3.7416573867739 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 = √300 - 13² - 9² - 7²

max_d = √300 - 169 - 81 - 49

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (13, 9, 7, 1) is an integer solution proven below

13² + 9² + 7² + 1² → 169 + 81 + 49 + 1 = 300

b = 10

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 13² - 10²)

max_c = Floor(√300 - 169 - 100)

max_c = Floor(√31)

max_c = Floor(5.56776436283)

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 → (300 - 13² - 10²)/2 = 15.5

When min_c = 4, then it is c² = 16 ≥ 15.5, so min_c = 4

c = 4

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

max_d = √300 - 13² - 10² - 4²

max_d = √300 - 169 - 100 - 16

max_d = √15

max_d = 3.8729833462074

Since max_d = 3.8729833462074 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 = √300 - 13² - 10² - 5²

max_d = √300 - 169 - 100 - 25

max_d = √6

max_d = 2.4494897427832

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

b = 11

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 13² - 11²)

max_c = Floor(√300 - 169 - 121)

max_c = Floor(√10)

max_c = Floor(3.1622776601684)

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 → (300 - 13² - 11²)/2 = 5

When min_c = 3, then it is c² = 9 ≥ 5, so min_c = 3

c = 3

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

max_d = √300 - 13² - 11² - 3²

max_d = √300 - 169 - 121 - 9

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (13, 11, 3, 1) is an integer solution proven below

13² + 11² + 3² + 1² → 169 + 121 + 9 + 1 = 300

a = 14

Find max_b which is Floor(√n - a²)

max_b = Floor(√300 - 14²)

max_b = Floor(√300 - 196)

max_b = Floor(√104)

max_b = Floor(10.198039027186)

max_b = 10

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 → (300 - 14²)/3 = 34.666666666667

When min_b = 6, then it is b² = 36 ≥ 34.666666666667, so min_b = 6

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

(6, 10)

b = 6

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 14² - 6²)

max_c = Floor(√300 - 196 - 36)

max_c = Floor(√68)

max_c = Floor(8.2462112512353)

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 → (300 - 14² - 6²)/2 = 34

When min_c = 6, then it is c² = 36 ≥ 34, so min_c = 6

c = 6

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

max_d = √300 - 14² - 6² - 6²

max_d = √300 - 196 - 36 - 36

max_d = √32

max_d = 5.6568542494924

Since max_d = 5.6568542494924 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 = √300 - 14² - 6² - 7²

max_d = √300 - 196 - 36 - 49

max_d = √19

max_d = 4.3588989435407

Since max_d = 4.3588989435407 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 = √300 - 14² - 6² - 8²

max_d = √300 - 196 - 36 - 64

max_d = √4

max_d = 2

Since max_d = 2, then (a, b, c, d) = (14, 6, 8, 2) is an integer solution proven below

14² + 6² + 8² + 2² → 196 + 36 + 64 + 4 = 300

b = 7

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 14² - 7²)

max_c = Floor(√300 - 196 - 49)

max_c = Floor(√55)

max_c = Floor(7.4161984870957)

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 → (300 - 14² - 7²)/2 = 27.5

When min_c = 6, then it is c² = 36 ≥ 27.5, so min_c = 6

c = 6

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

max_d = √300 - 14² - 7² - 6²

max_d = √300 - 196 - 49 - 36

max_d = √19

max_d = 4.3588989435407

Since max_d = 4.3588989435407 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 = √300 - 14² - 7² - 7²

max_d = √300 - 196 - 49 - 49

max_d = √6

max_d = 2.4494897427832

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

b = 8

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 14² - 8²)

max_c = Floor(√300 - 196 - 64)

max_c = Floor(√40)

max_c = Floor(6.3245553203368)

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 → (300 - 14² - 8²)/2 = 20

When min_c = 5, then it is c² = 25 ≥ 20, so min_c = 5

c = 5

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

max_d = √300 - 14² - 8² - 5²

max_d = √300 - 196 - 64 - 25

max_d = √15

max_d = 3.8729833462074

Since max_d = 3.8729833462074 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 = √300 - 14² - 8² - 6²

max_d = √300 - 196 - 64 - 36

max_d = √4

max_d = 2

Since max_d = 2, then (a, b, c, d) = (14, 8, 6, 2) is an integer solution proven below

14² + 8² + 6² + 2² → 196 + 64 + 36 + 4 = 300

b = 9

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 14² - 9²)

max_c = Floor(√300 - 196 - 81)

max_c = Floor(√23)

max_c = Floor(4.7958315233127)

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 → (300 - 14² - 9²)/2 = 11.5

When min_c = 4, then it is c² = 16 ≥ 11.5, so min_c = 4

c = 4

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

max_d = √300 - 14² - 9² - 4²

max_d = √300 - 196 - 81 - 16

max_d = √7

max_d = 2.6457513110646

Since max_d = 2.6457513110646 is not an integer, this is not a solution

b = 10

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 14² - 10²)

max_c = Floor(√300 - 196 - 100)

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 → (300 - 14² - 10²)/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 = √300 - 14² - 10² - 2²

max_d = √300 - 196 - 100 - 4

max_d = √0

max_d = 0

Since max_d = 0, then (a, b, c, d) = (14, 10, 2, 0) is an integer solution proven below

14² + 10² + 2² + 0² → 196 + 100 + 4 + 0 = 300

a = 15

Find max_b which is Floor(√n - a²)

max_b = Floor(√300 - 15²)

max_b = Floor(√300 - 225)

max_b = Floor(√75)

max_b = Floor(8.6602540378444)

max_b = 8

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 → (300 - 15²)/3 = 25

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

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

(5, 8)

b = 5

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 15² - 5²)

max_c = Floor(√300 - 225 - 25)

max_c = Floor(√50)

max_c = Floor(7.0710678118655)

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 → (300 - 15² - 5²)/2 = 25

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

c = 5

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

max_d = √300 - 15² - 5² - 5²

max_d = √300 - 225 - 25 - 25

max_d = √25

max_d = 5

Since max_d = 5, then (a, b, c, d) = (15, 5, 5, 5) is an integer solution proven below

15² + 5² + 5² + 5² → 225 + 25 + 25 + 25 = 300

c = 6

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

max_d = √300 - 15² - 5² - 6²

max_d = √300 - 225 - 25 - 36

max_d = √14

max_d = 3.7416573867739

Since max_d = 3.7416573867739 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 = √300 - 15² - 5² - 7²

max_d = √300 - 225 - 25 - 49

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (15, 5, 7, 1) is an integer solution proven below

15² + 5² + 7² + 1² → 225 + 25 + 49 + 1 = 300

b = 6

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 15² - 6²)

max_c = Floor(√300 - 225 - 36)

max_c = Floor(√39)

max_c = Floor(6.2449979983984)

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 → (300 - 15² - 6²)/2 = 19.5

When min_c = 5, then it is c² = 25 ≥ 19.5, so min_c = 5

c = 5

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

max_d = √300 - 15² - 6² - 5²

max_d = √300 - 225 - 36 - 25

max_d = √14

max_d = 3.7416573867739

Since max_d = 3.7416573867739 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 = √300 - 15² - 6² - 6²

max_d = √300 - 225 - 36 - 36

max_d = √3

max_d = 1.7320508075689

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

b = 7

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 15² - 7²)

max_c = Floor(√300 - 225 - 49)

max_c = Floor(√26)

max_c = Floor(5.0990195135928)

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 → (300 - 15² - 7²)/2 = 13

When min_c = 4, then it is c² = 16 ≥ 13, so min_c = 4

c = 4

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

max_d = √300 - 15² - 7² - 4²

max_d = √300 - 225 - 49 - 16

max_d = √10

max_d = 3.1622776601684

Since max_d = 3.1622776601684 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 = √300 - 15² - 7² - 5²

max_d = √300 - 225 - 49 - 25

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (15, 7, 5, 1) is an integer solution proven below

15² + 7² + 5² + 1² → 225 + 49 + 25 + 1 = 300

b = 8

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 15² - 8²)

max_c = Floor(√300 - 225 - 64)

max_c = Floor(√11)

max_c = Floor(3.3166247903554)

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 → (300 - 15² - 8²)/2 = 5.5

When min_c = 3, then it is c² = 9 ≥ 5.5, so min_c = 3

c = 3

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

max_d = √300 - 15² - 8² - 3²

max_d = √300 - 225 - 64 - 9

max_d = √2

max_d = 1.4142135623731

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

a = 16

Find max_b which is Floor(√n - a²)

max_b = Floor(√300 - 16²)

max_b = Floor(√300 - 256)

max_b = Floor(√44)

max_b = Floor(6.6332495807108)

max_b = 6

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 → (300 - 16²)/3 = 14.666666666667

When min_b = 4, then it is b² = 16 ≥ 14.666666666667, so min_b = 4

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

(4, 6)

b = 4

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 16² - 4²)

max_c = Floor(√300 - 256 - 16)

max_c = Floor(√28)

max_c = Floor(5.2915026221292)

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 → (300 - 16² - 4²)/2 = 14

When min_c = 4, then it is c² = 16 ≥ 14, so min_c = 4

c = 4

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

max_d = √300 - 16² - 4² - 4²

max_d = √300 - 256 - 16 - 16

max_d = √12

max_d = 3.4641016151378

Since max_d = 3.4641016151378 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 = √300 - 16² - 4² - 5²

max_d = √300 - 256 - 16 - 25

max_d = √3

max_d = 1.7320508075689

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

b = 5

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 16² - 5²)

max_c = Floor(√300 - 256 - 25)

max_c = Floor(√19)

max_c = Floor(4.3588989435407)

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 → (300 - 16² - 5²)/2 = 9.5

When min_c = 4, then it is c² = 16 ≥ 9.5, so min_c = 4

c = 4

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

max_d = √300 - 16² - 5² - 4²

max_d = √300 - 256 - 25 - 16

max_d = √3

max_d = 1.7320508075689

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

b = 6

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 16² - 6²)

max_c = Floor(√300 - 256 - 36)

max_c = Floor(√8)

max_c = Floor(2.8284271247462)

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 → (300 - 16² - 6²)/2 = 4

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

c = 2

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

max_d = √300 - 16² - 6² - 2²

max_d = √300 - 256 - 36 - 4

max_d = √4

max_d = 2

Since max_d = 2, then (a, b, c, d) = (16, 6, 2, 2) is an integer solution proven below

16² + 6² + 2² + 2² → 256 + 36 + 4 + 4 = 300

a = 17

Find max_b which is Floor(√n - a²)

max_b = Floor(√300 - 17²)

max_b = Floor(√300 - 289)

max_b = Floor(√11)

max_b = Floor(3.3166247903554)

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 → (300 - 17²)/3 = 3.6666666666667

When min_b = 2, then it is b² = 4 ≥ 3.6666666666667, so min_b = 2

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

(2, 3)

b = 2

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 17² - 2²)

max_c = Floor(√300 - 289 - 4)

max_c = Floor(√7)

max_c = Floor(2.6457513110646)

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 → (300 - 17² - 2²)/2 = 3.5

When min_c = 2, then it is c² = 4 ≥ 3.5, so min_c = 2

c = 2

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

max_d = √300 - 17² - 2² - 2²

max_d = √300 - 289 - 4 - 4

max_d = √3

max_d = 1.7320508075689

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

b = 3

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√300 - 17² - 3²)

max_c = Floor(√300 - 289 - 9)

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 → (300 - 17² - 3²)/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 = √300 - 17² - 3² - 1²

max_d = √300 - 289 - 9 - 1

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (17, 3, 1, 1) is an integer solution proven below

17² + 3² + 1² + 1² → 289 + 9 + 1 + 1 = 300

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Tags:

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