query: How to prove that all primes of the form $4k+1$ can be represented by the sum of two squares in only one way regardless of the order?
positive_passages.docid: 142584
positive_passages.text: Confusion in proof that primes $p = 4k + 1$ are uniquely the sums of two squares
negative_passages.docid: 146512
negative_passages.text: Prove that for every prime number $p>3$, $4p^2+1$ can be written as the sum of three square numbers
negative_passages.docid: 3632
negative_passages.text: Basic number theory proofs
negative_passages.docid: 40968
negative_passages.text: show if $n=4k+3$ is a prime and ${a^2+b^2} \equiv 0 \pmod n$ , then $a \equiv b \equiv 0 \pmod n$
negative_passages.docid: 70066
negative_passages.text: Show that product of primes, $\prod_{k=1}^{\pi(n)} p_k < 4^n$
negative_passages.docid: 39698
negative_passages.text: On uniqueness of sums of prime powers
negative_passages.docid: 111037
negative_passages.text: Consider numbers of the type $n=2^m+1$. Prove that such an n is prime only if $n=F_K$ for some $ k \in N$, where $F_k$ is a Fermat Prime.
negative_passages.docid: 81723
negative_passages.text: Infinitely many primes proof using Euler's totient
negative_passages.docid: 38498
negative_passages.text: Find all primes $p$ such that $(2^{p-1}-1)/p$ is a perfect square
negative_passages.docid: 44019
negative_passages.text: Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
negative_passages.docid: 16621
negative_passages.text: Help understand the proof of infinitely many primes of the form $4n+3$
negative_passages.docid: 106111
negative_passages.text: Understanding that there are infinitely many primes of form $4n+3$
negative_passages.docid: 67332
negative_passages.text: Sum of two squares modulo p
negative_passages.docid: 71110
negative_passages.text: Show that we cannot have a prime triplet of the form $p$, $p + 2$, $p + 4$ for $p >3$
negative_passages.docid: 98468
negative_passages.text: Are there infinitely many primes of the form $n!+1$?
negative_passages.docid: 100984
negative_passages.text: Prove that if prime $p$ divide $a_{2k}-2$, then $p$ divide also $a_{2k+1}-1$.
negative_passages.docid: 136947
negative_passages.text: Simple Mersenne prime divisibility proofs