Lagrange four-square (Bachet Conjecture) for 91

Crop Image

×

Crop

Enter Number

Show the Lagrange Four Square Theorem for

91

Lagrange Four Square Definition

For any natural number (p), we write as

p = a² + b² + c² + d²

Determine max_a:

Floor(√91) = Floor(9.5393920141695)

Floor(9.5393920141695) = 9
This is called max_a

Determine min_a:

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

Start with min_a = 0 and increase by 1

Continue until we reach or breach n/4 → 91/4 = 22.75

When min_a = 5, then it is a² = 25 ≥ 22.75, so min_a = 5

Find a in the range of (min_a, max_a)

(0, 9)

a = 0

Find max_b which is Floor(√n - a²)

max_b = Floor(√91 - 0²)

max_b = Floor(√91 - 0)

max_b = Floor(√91)

max_b = Floor(9.5393920141695)

max_b = 9

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 0 and increase by 1
Go until (n - a²)/3 → (91 - 0²)/3 = 30.333333333333

When min_b = 6, then it is b² = 36 ≥ 30.333333333333, 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(√91 - 0² - 6²)

max_c = Floor(√91 - 0 - 36)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 0² - 6²)/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 = √91 - 0² - 6² - 6²

max_d = √91 - 0 - 36 - 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 = √91 - 0² - 6² - 7²

max_d = √91 - 0 - 36 - 49

max_d = √6

max_d = 2.4494897427832

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

b = 7

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 0² - 7²)

max_c = Floor(√91 - 0 - 49)

max_c = Floor(√42)

max_c = Floor(6.4807406984079)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 0² - 7²)/2 = 21

When min_c = 5, then it is c² = 25 ≥ 21, so min_c = 5

c = 5

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

max_d = √91 - 0² - 7² - 5²

max_d = √91 - 0 - 49 - 25

max_d = √17

max_d = 4.1231056256177

Since max_d = 4.1231056256177 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 = √91 - 0² - 7² - 6²

max_d = √91 - 0 - 49 - 36

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(√91 - 0² - 8²)

max_c = Floor(√91 - 0 - 64)

max_c = Floor(√27)

max_c = Floor(5.1961524227066)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 0² - 8²)/2 = 13.5

When min_c = 4, then it is c² = 16 ≥ 13.5, so min_c = 4

c = 4

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

max_d = √91 - 0² - 8² - 4²

max_d = √91 - 0 - 64 - 16

max_d = √11

max_d = 3.3166247903554

Since max_d = 3.3166247903554 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 = √91 - 0² - 8² - 5²

max_d = √91 - 0 - 64 - 25

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(√91 - 0² - 9²)

max_c = Floor(√91 - 0 - 81)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 0² - 9²)/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 = √91 - 0² - 9² - 3²

max_d = √91 - 0 - 81 - 9

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (0, 9, 3, 1) is an integer solution proven below

0² + 9² + 3² + 1² → 0 + 81 + 9 + 1 = 91

a = 1

Find max_b which is Floor(√n - a²)

max_b = Floor(√91 - 1²)

max_b = Floor(√91 - 1)

max_b = Floor(√90)

max_b = Floor(9.4868329805051)

max_b = 9

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 0 and increase by 1
Go until (n - a²)/3 → (91 - 1²)/3 = 30

When min_b = 6, then it is b² = 36 ≥ 30, 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(√91 - 1² - 6²)

max_c = Floor(√91 - 1 - 36)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 1² - 6²)/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 = √91 - 1² - 6² - 6²

max_d = √91 - 1 - 36 - 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 = √91 - 1² - 6² - 7²

max_d = √91 - 1 - 36 - 49

max_d = √5

max_d = 2.2360679774998

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

b = 7

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 1² - 7²)

max_c = Floor(√91 - 1 - 49)

max_c = Floor(√41)

max_c = Floor(6.4031242374328)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 1² - 7²)/2 = 20.5

When min_c = 5, then it is c² = 25 ≥ 20.5, so min_c = 5

c = 5

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

max_d = √91 - 1² - 7² - 5²

max_d = √91 - 1 - 49 - 25

max_d = √16

max_d = 4

Since max_d = 4, then (a, b, c, d) = (1, 7, 5, 4) is an integer solution proven below

1² + 7² + 5² + 4² → 1 + 49 + 25 + 16 = 91

c = 6

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

max_d = √91 - 1² - 7² - 6²

max_d = √91 - 1 - 49 - 36

max_d = √5

max_d = 2.2360679774998

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

b = 8

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 1² - 8²)

max_c = Floor(√91 - 1 - 64)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 1² - 8²)/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 = √91 - 1² - 8² - 4²

max_d = √91 - 1 - 64 - 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 = √91 - 1² - 8² - 5²

max_d = √91 - 1 - 64 - 25

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (1, 8, 5, 1) is an integer solution proven below

1² + 8² + 5² + 1² → 1 + 64 + 25 + 1 = 91

b = 9

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 1² - 9²)

max_c = Floor(√91 - 1 - 81)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 1² - 9²)/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 = √91 - 1² - 9² - 3²

max_d = √91 - 1 - 81 - 9

max_d = √0

max_d = 0

Since max_d = 0, then (a, b, c, d) = (1, 9, 3, 0) is an integer solution proven below

1² + 9² + 3² + 0² → 1 + 81 + 9 + 0 = 91

a = 2

Find max_b which is Floor(√n - a²)

max_b = Floor(√91 - 2²)

max_b = Floor(√91 - 4)

max_b = Floor(√87)

max_b = Floor(9.3273790530888)

max_b = 9

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 0 and increase by 1
Go until (n - a²)/3 → (91 - 2²)/3 = 29

When min_b = 6, then it is b² = 36 ≥ 29, 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(√91 - 2² - 6²)

max_c = Floor(√91 - 4 - 36)

max_c = Floor(√51)

max_c = Floor(7.1414284285429)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 2² - 6²)/2 = 25.5

When min_c = 6, then it is c² = 36 ≥ 25.5, so min_c = 6

c = 6

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

max_d = √91 - 2² - 6² - 6²

max_d = √91 - 4 - 36 - 36

max_d = √15

max_d = 3.8729833462074

Since max_d = 3.8729833462074 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 = √91 - 2² - 6² - 7²

max_d = √91 - 4 - 36 - 49

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(√91 - 2² - 7²)

max_c = Floor(√91 - 4 - 49)

max_c = Floor(√38)

max_c = Floor(6.164414002969)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 2² - 7²)/2 = 19

When min_c = 5, then it is c² = 25 ≥ 19, so min_c = 5

c = 5

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

max_d = √91 - 2² - 7² - 5²

max_d = √91 - 4 - 49 - 25

max_d = √13

max_d = 3.605551275464

Since max_d = 3.605551275464 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 = √91 - 2² - 7² - 6²

max_d = √91 - 4 - 49 - 36

max_d = √2

max_d = 1.4142135623731

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

b = 8

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 2² - 8²)

max_c = Floor(√91 - 4 - 64)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 2² - 8²)/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 = √91 - 2² - 8² - 4²

max_d = √91 - 4 - 64 - 16

max_d = √7

max_d = 2.6457513110646

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

b = 9

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 2² - 9²)

max_c = Floor(√91 - 4 - 81)

max_c = Floor(√6)

max_c = Floor(2.4494897427832)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 2² - 9²)/2 = 3

When min_c = 2, then it is c² = 4 ≥ 3, so min_c = 2

c = 2

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

max_d = √91 - 2² - 9² - 2²

max_d = √91 - 4 - 81 - 4

max_d = √2

max_d = 1.4142135623731

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

a = 3

Find max_b which is Floor(√n - a²)

max_b = Floor(√91 - 3²)

max_b = Floor(√91 - 9)

max_b = Floor(√82)

max_b = Floor(9.0553851381374)

max_b = 9

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 0 and increase by 1
Go until (n - a²)/3 → (91 - 3²)/3 = 27.333333333333

When min_b = 6, then it is b² = 36 ≥ 27.333333333333, 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(√91 - 3² - 6²)

max_c = Floor(√91 - 9 - 36)

max_c = Floor(√46)

max_c = Floor(6.7823299831253)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 3² - 6²)/2 = 23

When min_c = 5, then it is c² = 25 ≥ 23, so min_c = 5

c = 5

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

max_d = √91 - 3² - 6² - 5²

max_d = √91 - 9 - 36 - 25

max_d = √21

max_d = 4.5825756949558

Since max_d = 4.5825756949558 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 = √91 - 3² - 6² - 6²

max_d = √91 - 9 - 36 - 36

max_d = √10

max_d = 3.1622776601684

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

b = 7

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 3² - 7²)

max_c = Floor(√91 - 9 - 49)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 3² - 7²)/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 = √91 - 3² - 7² - 5²

max_d = √91 - 9 - 49 - 25

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(√91 - 3² - 8²)

max_c = Floor(√91 - 9 - 64)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 3² - 8²)/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 = √91 - 3² - 8² - 3²

max_d = √91 - 9 - 64 - 9

max_d = √9

max_d = 3

Since max_d = 3, then (a, b, c, d) = (3, 8, 3, 3) is an integer solution proven below

3² + 8² + 3² + 3² → 9 + 64 + 9 + 9 = 91

c = 4

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

max_d = √91 - 3² - 8² - 4²

max_d = √91 - 9 - 64 - 16

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(√91 - 3² - 9²)

max_c = Floor(√91 - 9 - 81)

max_c = Floor(√1)

max_c = Floor(1)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 3² - 9²)/2 = 0.5

When min_c = 0, then it is c² = 1 ≥ 0.5, so min_c = 0

c = 0

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

max_d = √91 - 3² - 9² - 0²

max_d = √91 - 9 - 81 - 0

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (3, 9, 0, 1) is an integer solution proven below

3² + 9² + 0² + 1² → 9 + 81 + 0 + 1 = 91

c = 1

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

max_d = √91 - 3² - 9² - 1²

max_d = √91 - 9 - 81 - 1

max_d = √0

max_d = 0

Since max_d = 0, then (a, b, c, d) = (3, 9, 1, 0) is an integer solution proven below

3² + 9² + 1² + 0² → 9 + 81 + 1 + 0 = 91

a = 4

Find max_b which is Floor(√n - a²)

max_b = Floor(√91 - 4²)

max_b = Floor(√91 - 16)

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 = 0 and increase by 1
Go until (n - a²)/3 → (91 - 4²)/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(√91 - 4² - 5²)

max_c = Floor(√91 - 16 - 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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 4² - 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 = √91 - 4² - 5² - 5²

max_d = √91 - 16 - 25 - 25

max_d = √25

max_d = 5

Since max_d = 5, then (a, b, c, d) = (4, 5, 5, 5) is an integer solution proven below

4² + 5² + 5² + 5² → 16 + 25 + 25 + 25 = 91

c = 6

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

max_d = √91 - 4² - 5² - 6²

max_d = √91 - 16 - 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 = √91 - 4² - 5² - 7²

max_d = √91 - 16 - 25 - 49

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (4, 5, 7, 1) is an integer solution proven below

4² + 5² + 7² + 1² → 16 + 25 + 49 + 1 = 91

b = 6

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 4² - 6²)

max_c = Floor(√91 - 16 - 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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 4² - 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 = √91 - 4² - 6² - 5²

max_d = √91 - 16 - 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 = √91 - 4² - 6² - 6²

max_d = √91 - 16 - 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(√91 - 4² - 7²)

max_c = Floor(√91 - 16 - 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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 4² - 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 = √91 - 4² - 7² - 4²

max_d = √91 - 16 - 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 = √91 - 4² - 7² - 5²

max_d = √91 - 16 - 49 - 25

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (4, 7, 5, 1) is an integer solution proven below

4² + 7² + 5² + 1² → 16 + 49 + 25 + 1 = 91

b = 8

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 4² - 8²)

max_c = Floor(√91 - 16 - 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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 4² - 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 = √91 - 4² - 8² - 3²

max_d = √91 - 16 - 64 - 9

max_d = √2

max_d = 1.4142135623731

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

a = 5

Find max_b which is Floor(√n - a²)

max_b = Floor(√91 - 5²)

max_b = Floor(√91 - 25)

max_b = Floor(√66)

max_b = Floor(8.124038404636)

max_b = 8

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 0 and increase by 1
Go until (n - a²)/3 → (91 - 5²)/3 = 22

When min_b = 5, then it is b² = 25 ≥ 22, 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(√91 - 5² - 5²)

max_c = Floor(√91 - 25 - 25)

max_c = Floor(√41)

max_c = Floor(6.4031242374328)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 5² - 5²)/2 = 20.5

When min_c = 5, then it is c² = 25 ≥ 20.5, so min_c = 5

c = 5

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

max_d = √91 - 5² - 5² - 5²

max_d = √91 - 25 - 25 - 25

max_d = √16

max_d = 4

Since max_d = 4, then (a, b, c, d) = (5, 5, 5, 4) is an integer solution proven below

5² + 5² + 5² + 4² → 25 + 25 + 25 + 16 = 91

c = 6

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

max_d = √91 - 5² - 5² - 6²

max_d = √91 - 25 - 25 - 36

max_d = √5

max_d = 2.2360679774998

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

b = 6

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 5² - 6²)

max_c = Floor(√91 - 25 - 36)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 5² - 6²)/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 = √91 - 5² - 6² - 4²

max_d = √91 - 25 - 36 - 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 = √91 - 5² - 6² - 5²

max_d = √91 - 25 - 36 - 25

max_d = √5

max_d = 2.2360679774998

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

b = 7

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 5² - 7²)

max_c = Floor(√91 - 25 - 49)

max_c = Floor(√17)

max_c = Floor(4.1231056256177)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 5² - 7²)/2 = 8.5

When min_c = 3, then it is c² = 9 ≥ 8.5, so min_c = 3

c = 3

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

max_d = √91 - 5² - 7² - 3²

max_d = √91 - 25 - 49 - 9

max_d = √8

max_d = 2.8284271247462

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

c = 4

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

max_d = √91 - 5² - 7² - 4²

max_d = √91 - 25 - 49 - 16

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (5, 7, 4, 1) is an integer solution proven below

5² + 7² + 4² + 1² → 25 + 49 + 16 + 1 = 91

b = 8

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 5² - 8²)

max_c = Floor(√91 - 25 - 64)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 5² - 8²)/2 = 1

When min_c = 0, then it is c² = 1 ≥ 1, so min_c = 0

c = 0

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

max_d = √91 - 5² - 8² - 0²

max_d = √91 - 25 - 64 - 0

max_d = √2

max_d = 1.4142135623731

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

c = 1

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

max_d = √91 - 5² - 8² - 1²

max_d = √91 - 25 - 64 - 1

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (5, 8, 1, 1) is an integer solution proven below

5² + 8² + 1² + 1² → 25 + 64 + 1 + 1 = 91

a = 6

Find max_b which is Floor(√n - a²)

max_b = Floor(√91 - 6²)

max_b = Floor(√91 - 36)

max_b = Floor(√55)

max_b = Floor(7.4161984870957)

max_b = 7

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 0 and increase by 1
Go until (n - a²)/3 → (91 - 6²)/3 = 18.333333333333

When min_b = 5, then it is b² = 25 ≥ 18.333333333333, 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(√91 - 6² - 5²)

max_c = Floor(√91 - 36 - 25)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 6² - 5²)/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 = √91 - 6² - 5² - 4²

max_d = √91 - 36 - 25 - 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 = √91 - 6² - 5² - 5²

max_d = √91 - 36 - 25 - 25

max_d = √5

max_d = 2.2360679774998

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

b = 6

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 6² - 6²)

max_c = Floor(√91 - 36 - 36)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 6² - 6²)/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 = √91 - 6² - 6² - 4²

max_d = √91 - 36 - 36 - 16

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(√91 - 6² - 7²)

max_c = Floor(√91 - 36 - 49)

max_c = Floor(√6)

max_c = Floor(2.4494897427832)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 6² - 7²)/2 = 3

When min_c = 2, then it is c² = 4 ≥ 3, so min_c = 2

c = 2

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

max_d = √91 - 6² - 7² - 2²

max_d = √91 - 36 - 49 - 4

max_d = √2

max_d = 1.4142135623731

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

a = 7

Find max_b which is Floor(√n - a²)

max_b = Floor(√91 - 7²)

max_b = Floor(√91 - 49)

max_b = Floor(√42)

max_b = Floor(6.4807406984079)

max_b = 6

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 0 and increase by 1
Go until (n - a²)/3 → (91 - 7²)/3 = 14

When min_b = 4, then it is b² = 16 ≥ 14, 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(√91 - 7² - 4²)

max_c = Floor(√91 - 49 - 16)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 7² - 4²)/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 = √91 - 7² - 4² - 4²

max_d = √91 - 49 - 16 - 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 = √91 - 7² - 4² - 5²

max_d = √91 - 49 - 16 - 25

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (7, 4, 5, 1) is an integer solution proven below

7² + 4² + 5² + 1² → 49 + 16 + 25 + 1 = 91

b = 5

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 7² - 5²)

max_c = Floor(√91 - 49 - 25)

max_c = Floor(√17)

max_c = Floor(4.1231056256177)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 7² - 5²)/2 = 8.5

When min_c = 3, then it is c² = 9 ≥ 8.5, so min_c = 3

c = 3

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

max_d = √91 - 7² - 5² - 3²

max_d = √91 - 49 - 25 - 9

max_d = √8

max_d = 2.8284271247462

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

c = 4

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

max_d = √91 - 7² - 5² - 4²

max_d = √91 - 49 - 25 - 16

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (7, 5, 4, 1) is an integer solution proven below

7² + 5² + 4² + 1² → 49 + 25 + 16 + 1 = 91

b = 6

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 7² - 6²)

max_c = Floor(√91 - 49 - 36)

max_c = Floor(√6)

max_c = Floor(2.4494897427832)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 7² - 6²)/2 = 3

When min_c = 2, then it is c² = 4 ≥ 3, so min_c = 2

c = 2

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

max_d = √91 - 7² - 6² - 2²

max_d = √91 - 49 - 36 - 4

max_d = √2

max_d = 1.4142135623731

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

a = 8

Find max_b which is Floor(√n - a²)

max_b = Floor(√91 - 8²)

max_b = Floor(√91 - 64)

max_b = Floor(√27)

max_b = Floor(5.1961524227066)

max_b = 5

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 0 and increase by 1
Go until (n - a²)/3 → (91 - 8²)/3 = 9

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

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

(3, 5)

b = 3

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 8² - 3²)

max_c = Floor(√91 - 64 - 9)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 8² - 3²)/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 = √91 - 8² - 3² - 3²

max_d = √91 - 64 - 9 - 9

max_d = √9

max_d = 3

Since max_d = 3, then (a, b, c, d) = (8, 3, 3, 3) is an integer solution proven below

8² + 3² + 3² + 3² → 64 + 9 + 9 + 9 = 91

c = 4

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

max_d = √91 - 8² - 3² - 4²

max_d = √91 - 64 - 9 - 16

max_d = √2

max_d = 1.4142135623731

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

b = 4

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 8² - 4²)

max_c = Floor(√91 - 64 - 16)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 8² - 4²)/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 = √91 - 8² - 4² - 3²

max_d = √91 - 64 - 16 - 9

max_d = √2

max_d = 1.4142135623731

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

b = 5

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 8² - 5²)

max_c = Floor(√91 - 64 - 25)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 8² - 5²)/2 = 1

When min_c = 0, then it is c² = 1 ≥ 1, so min_c = 0

c = 0

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

max_d = √91 - 8² - 5² - 0²

max_d = √91 - 64 - 25 - 0

max_d = √2

max_d = 1.4142135623731

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

c = 1

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

max_d = √91 - 8² - 5² - 1²

max_d = √91 - 64 - 25 - 1

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (8, 5, 1, 1) is an integer solution proven below

8² + 5² + 1² + 1² → 64 + 25 + 1 + 1 = 91

a = 9

Find max_b which is Floor(√n - a²)

max_b = Floor(√91 - 9²)

max_b = Floor(√91 - 81)

max_b = Floor(√10)

max_b = Floor(3.1622776601684)

max_b = 3

Find b such that b² ≥ (n - a²)/3

Call it min_b

Find b

Start with min_b = 0 and increase by 1
Go until (n - a²)/3 → (91 - 9²)/3 = 3.3333333333333

When min_b = 2, then it is b² = 4 ≥ 3.3333333333333, 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(√91 - 9² - 2²)

max_c = Floor(√91 - 81 - 4)

max_c = Floor(√6)

max_c = Floor(2.4494897427832)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 9² - 2²)/2 = 3

When min_c = 2, then it is c² = 4 ≥ 3, so min_c = 2

c = 2

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

max_d = √91 - 9² - 2² - 2²

max_d = √91 - 81 - 4 - 4

max_d = √2

max_d = 1.4142135623731

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

b = 3

Determine max_c =Floor(√n - a² - b²)

max_c = Floor(√91 - 9² - 3²)

max_c = Floor(√91 - 81 - 9)

max_c = Floor(√1)

max_c = Floor(1)

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 = 0 and increase by 1

Go until (n - a² - b² )/2 → (91 - 9² - 3²)/2 = 0.5

When min_c = 0, then it is c² = 1 ≥ 0.5, so min_c = 0

c = 0

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

max_d = √91 - 9² - 3² - 0²

max_d = √91 - 81 - 9 - 0

max_d = √1

max_d = 1

Since max_d = 1, then (a, b, c, d) = (9, 3, 0, 1) is an integer solution proven below

9² + 3² + 0² + 1² → 81 + 9 + 0 + 1 = 91

c = 1

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

max_d = √91 - 9² - 3² - 1²

max_d = √91 - 81 - 9 - 1

max_d = √0

max_d = 0

Since max_d = 0, then (a, b, c, d) = (9, 3, 1, 0) is an integer solution proven below

9² + 3² + 1² + 0² → 81 + 9 + 1 + 0 = 91

List out 5 solutions:

You have 2 free calculationss remaining

Lagrange Four Square Theorem (Bachet Conjecture) Calculator Video

Tags:

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