query: Prove that every prime except one has the form $a^2 − b^2$ for some positive integers $a$ and $b$
positive_passages.docid: 18633
positive_passages.text: Prove every odd integer is the difference of two squares
negative_passages.docid: 75206
negative_passages.text: If $a$, $a+2$ and $a+4$ are prime numbers then, how can one prove that there is only one solution for $a$?
negative_passages.docid: 71880
negative_passages.text: Prove that every prime factor of $12x^2+1$ has the form $6n+1$
negative_passages.docid: 34323
negative_passages.text: Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$
negative_passages.docid: 48691
negative_passages.text: Product of all prime numbers upto some prime $p$
negative_passages.docid: 141102
negative_passages.text: If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$.
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: 72672
negative_passages.text: Show that if $ar + bs = 1$ for some $r$ and $s$ then $a$ and $b$ are relatively prime
negative_passages.docid: 140294
negative_passages.text: prove that $n >2$ and $n$ is a prime then $n$ is odd
negative_passages.docid: 34692
negative_passages.text: Prove if $ p$ is a prime of the form $3n+1$, then $p$ is also of the form $6m+1$
negative_passages.docid: 50663
negative_passages.text: prove or disprove: For a prime number p, there is a positive integer n such that $p+1 $= $n^2$ if and only if $ p = 3.$
negative_passages.docid: 32762
negative_passages.text: Proof for: $(a+b)^{p} \equiv a^p + b^p \pmod p$
negative_passages.docid: 21383
negative_passages.text: Can every even integer be expressed as the difference of two primes?
negative_passages.docid: 108237
negative_passages.text: Show $a^p \equiv b^p \mod p^2$
negative_passages.docid: 158844
negative_passages.text: If $a$ and $b$ are positive real numbers such that $a+b=1$, prove that $(a+1/a)^2+(b+1/b)^2\ge 25/2$
negative_passages.docid: 157131
negative_passages.text: Show that all Fermat numbers pass the base $2$ test (pseudoprime).
negative_passages.docid: 7491
negative_passages.text: How to show that $7\mid a^2+b^2$ implies $7\mid a$ and $7\mid b$?