query: If $p$ and $q$ are distinct primes then for any integer $a, pq\mid (a^{pq}-a^p-a^q+a)$
positive_passages.docid: 64148
positive_passages.text: If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$.
negative_passages.docid: 147777
negative_passages.text: If $P$ and $Q$ are distinct primes, how to prove that $\sqrt{PQ}$ is irrational?
negative_passages.docid: 135682
negative_passages.text: Are distinct prime ideals in a ring always coprime? If not, then when are they?
negative_passages.docid: 31917
negative_passages.text: If $q$ is coprime to $a$ then $a\mid nq-1,\,$ so $q$ is invertible mod $a$
negative_passages.docid: 11564
negative_passages.text: $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})=\mathbb{Q}(\sqrt{p}\sqrt[3]{q})$ for distinct prime $p,q$
negative_passages.docid: 64782
negative_passages.text: How to prove if $n$ is prime and $n | a^2$ then $n | a$?
negative_passages.docid: 156272
negative_passages.text: Unique pair of positive integers $(p,n)$ satisfying $p^3-p=n^7-n^3$ where $p$ is prime
negative_passages.docid: 5978
negative_passages.text: Prove that there exists only one prime number of the form $p^2−1$ where $p≥2$ is an integer
negative_passages.docid: 22742
negative_passages.text: Prove that if $p$ and $p^2+2$ are prime then $p^3+2$ is prime too
negative_passages.docid: 147101
negative_passages.text: Show that if $p>5$ is prime, then $240 \mid (p^4-1)$
negative_passages.docid: 140830
negative_passages.text: Show $\,p\mid 9^q -1 \implies q\mid p-1$ for odd primes $p$ and $q$
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: 33714
negative_passages.text: $p$ prime, $p\mid a^k \Rightarrow p^k\mid a^k$
negative_passages.docid: 83407
negative_passages.text: How to show that that $q \mid p-1$?
negative_passages.docid: 129061
negative_passages.text: How to say that any group of order $pqr$ is cyclic with a provided relation between $p$, $q$, $r$, where they are all distinct primes.
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: 103496
negative_passages.text: Show that $3p^2=q^2$ implies $3|p$ and $3|q$