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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?

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

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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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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