I have been wondering how to solve certain problem types, but specifically this one:

"If $p$ is a prime that can be written in the form $5x^2+6y^2$, where $x$ and $y$ are positive integers, prove that this representation is unique (i.e. no other $x$ and $y$ can create $p$)."

What would be a rigorous proof of this claim, and the main method used?


closed as too broad by Saad, TheSimpliFire, A. Goodier, Parcly Taxel, user223391 Feb 28 '18 at 13:50

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Theorem: Let $N>1$ be an odd integer expressed in two different ways as $$ N = m a^2 + n b^2 = m c^2 + n d^2, $$ where $a,b,c,d,m,n$ are positive integers, $b < d,$ $\gcd(ma, nb) = \gcd(mc,nd) = 1.$ Then $$ N = \gcd(N, ad-bc) \; \cdot \; \frac{N}{\gcd(N, ad-bc)} $$ gives a factorization, as both factors are larger than $1.$

From A Note on Euler's Factoring Problem by John Brillhart, The American Mathematical Monthly, volume 116, December 2009, pages 928-931. He also points out the surprising fact that the given factorization makes no explicit use of the numbers $m,n$


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