query: Show that if $p>2 $ is prime, then $8$ divides $p^2 -1$.
positive_passages.docid: 15915
positive_passages.text: For any prime $p > 3$, why is $p^2-1$ always divisible by 24?
negative_passages.docid: 32618
negative_passages.text: Suppose $p$ is a prime number such that $(p-1)/4$ and $(p+1)/2$ are also primes. Show that $p=13$
negative_passages.docid: 20881
negative_passages.text: Prove that if $p$ and $q$ are distinct primes such that $pq\mid n^2$ then $pq\mid n$.
negative_passages.docid: 147101
negative_passages.text: Show that if $p>5$ is prime, then $240 \mid (p^4-1)$
negative_passages.docid: 63893
negative_passages.text: Prime dividing the binomial coefficients
negative_passages.docid: 66703
negative_passages.text: Show that $8 \mid (a^2-b^2)$ for $a$ and $b$ both odd
negative_passages.docid: 145041
negative_passages.text: Given that $p$ is a prime and $p\mid a^n$, prove that $p^n\mid a^n$.
negative_passages.docid: 50894
negative_passages.text: Prove that $6$ divides $n(n + 1)(n + 2)$
negative_passages.docid: 126893
negative_passages.text: A group of order 2p (p prime) and other conditions - prove abelian.
negative_passages.docid: 26503
negative_passages.text: Suppose that $p$ ≥ $q$ ≥ $5$ are both prime numbers. Prove that 24 divides ($p^2 − q^2$)
negative_passages.docid: 140294
negative_passages.text: prove that $n >2$ and $n$ is a prime then $n$ is odd
negative_passages.docid: 77806
negative_passages.text: Proving if $a$ is not a multiple of prime $p$, then an integer $b$ exists such that $p^b - 1$ is a multiple of $a$
negative_passages.docid: 86594
negative_passages.text: If $x^2 \equiv y^2 \pmod {p^r}$, where $p$ is an odd prime not dividing $x$ or $y$ then $x \equiv \pm y \pmod {p^r}$
negative_passages.docid: 59036
negative_passages.text: If $a^{n}-1$ is prime then $a=2$ and $n$ is prime?
negative_passages.docid: 81601
negative_passages.text: Prove that if $x$ is odd then $x^2 -1$ is divisible by $8$.
negative_passages.docid: 50596
negative_passages.text: $p_1p_2...p_n +1$ is not square for $p_k = k$'th prime
negative_passages.docid: 143772
negative_passages.text: How can I algebraically prove that $2^n - 1$ is not always prime?