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Thue's lemma (Theorem)

Let $ p$ be a prime number of the form $ 4k+1$ . Then there are two unique integers $ a$ and $ b$ with $ 0<a<b$ such that $ p=a^2+b^2$. Additionally, if a number $ p$ can be written in as the sum of two squares in 2 different ways (i.e. $ p=a^2+b^2$ and $ p=c^2+d^2$ with the two sums being different), then the number $ p$ is composite.



"Thue's lemma" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: representing primes as $x^2+ny^2$

Keywords:  prime number, sum of squares

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proof of Thue's Lemma (Proof) by mathcam
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Cross-references: composite, sums, sum of two squares, number, integers, prime number
There are 2 references to this entry.

This is version 5 of Thue's lemma, born on 2002-12-25, modified 2005-03-18.
Object id is 3826, canonical name is ThuesLemma2.
Accessed 2953 times total.

Classification:
AMS MSC: 11A41 (Number theory :: Elementary number theory :: Primes)
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