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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?
2 Answers
- JLv 74 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.
Sorry, you'll have to google.
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- Ian HLv 74 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
