In how many ways can we solve

for

if is an odd prime?

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In how many ways can we solve

for

if is an odd prime?

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p^2 = a^2 + b^2 + c^2 for 0 < a <= b <= c

Some solutions for the problem are: [given here only distinct a,b,c]

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

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

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

17^2 = 8^2 + 9^2 + 12^2

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

19^2 = 6^2 + 10^2 + 15^2

Write p as p = 8*n ± 1 or p = 8*n ± 5

Then it has n solutions

The proof is based on known results involving ternary forms.