Ian H
Lv 7
Ian H asked in Science & MathematicsMathematics · 3 years ago

# Other identities for sum of 3 squares itself a square? 1 ^2 + 2 ^2 + 2 ^2 = 3 ^2 is from x^2 + (x + 1)^2 + (x^2 + x)^2 = (x^2 + x + 1)^2?

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• J
Lv 7
3 years ago

Closely related is this identity that I discovered in 2012

Let:

A = 4*a^15 + 10*a^12 + 28*a^9 + 32*a^6 + 8*a^3 - 1

B = a^16 - 8*a^13 - 32*a^10 - 28*a^7 - 10*a^4 - 4*a

C = a^16 + 13*a^13 + 10*a^10 - 10*a^7 - 13*a^4 - a

D = a^15 + 13*a^12 + 10*a^9 - 10*a^6 - 13*a^3 - 1

then:

A^3 + B^3 = C^3 + D^3

many identities, including for squares, are at Tito Piezas web site.

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• Ian H
Lv 7
3 years ago

Extra note:

Could lists like this be predicted other than by trial and error ?

1 ^2 + 4 ^2 + 8 ^2 = 9 ^2

1 ^2 + 6 ^2 + 18 ^2 = 19 ^2

1 ^2 + 12 ^2 + 12 ^2 = 17 ^2

2 ^2 + 3 ^2 + 6 ^2 = 7 ^2

2 ^2 + 4 ^2 + 4 ^2 = 6 ^2

2 ^2 + 5 ^2 + 14 ^2 = 15 ^2

2 ^2 + 6 ^2 + 9 ^2 = 11 ^2

2 ^2 + 7 ^2 + 26 ^2 = 27 ^2

2 ^2 + 8 ^2 + 16 ^2 = 18 ^2

2 ^2 + 10 ^2 + 11 ^2 = 15 ^2

2 ^2 + 10 ^2 + 25 ^2 = 27 ^2

2 ^2 + 14 ^2 + 23 ^2 = 27 ^2

2 ^2 + 24 ^2 + 24 ^2 = 34 ^2

2 ^2 + 26 ^2 + 29 ^2 = 39 ^2

3 ^2 + 4 ^2 + 12 ^2 = 13 ^2

3 ^2 + 6 ^2 + 6 ^2 = 9 ^2

3 ^2 + 6 ^2 + 22 ^2 = 23 ^2

3 ^2 + 12 ^2 + 24 ^2 = 27 ^2

3 ^2 + 14 ^2 + 18 ^2 = 23 ^2

3 ^2 + 16 ^2 + 24 ^2 = 29 ^2

3 ^2 + 24 ^2 + 28 ^2 = 37 ^2

4 ^2 + 4 ^2 + 7 ^2 = 9 ^2

4 ^2 + 5 ^2 + 20 ^2 = 21 ^2

4 ^2 + 6 ^2 + 12 ^2 = 14 ^2

4 ^2 + 8 ^2 + 8 ^2 = 12 ^2

4 ^2 + 8 ^2 + 19 ^2 = 21 ^2

4 ^2 + 10 ^2 + 28 ^2 = 30 ^2

4 ^2 + 12 ^2 + 18 ^2 = 22 ^2

4 ^2 + 13 ^2 + 16 ^2 = 21 ^2

4 ^2 + 17 ^2 + 28 ^2 = 33 ^2

4 ^2 + 20 ^2 + 22 ^2 = 30 ^2

5 ^2 + 10 ^2 + 10 ^2 = 15 ^2