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True or false: An integer prime p can be written as a sum of two squares if and only if ...?
True or false:
An integer prime p can be written as the sum of two squares if and only if there exists an integer n such that n^2 is congruent to -1 (modulo p).
Prove your answer in a fairly elementary way.
I'm not sure I understood that infinite-regress proof, actually.
2 Answers
- Anonymous1 decade agoFavorite Answer
It's true.
Consider first n^2 + 1 = k.p where n > k > 1
Let n = i.k + m
Prove that m^2 + 1 = k.j where j < k
Then prove that you can cancel out k^2 from
(n^2 + 1)(m^2 + 1) = j.k^2.p
to leave r^2 + s^2 = j.p . . . and if j > 1, you can reduce both r and s modulo j to construct t^2 + u^2 = j.h where h < j, and multiply out to cancel j^2 . . . so you can keep going, and the only thing that can stop you is x^2 + y^2 = L.n when L = 1.
For the other way round, start with a^2 + b^2 = p, and multiply by c^2 chosen so that b.c = k.p + 1.
Elementary number theory is FUN!
Source(s): "God made the integers, all else is the work of man" - Leopold Kronecker (1823-1891).
