query: Prove that $a|p+1$ if and only if ${\frac{a}{p} = \frac{1}{n} + \frac{1}{m} }$ such that $n,m \in N$ and $p$ is prime
positive_passages.docid: 27170
positive_passages.text: $a\in \mathbb{N}$, $p$ prime, $a<p$ prove that $a\mid p+1\iff\exists\, b,c\in\Bbb N:\dfrac{a}{p}=\dfrac{1}{b}+\dfrac{1}{c}$
negative_passages.docid: 65511
negative_passages.text: Proving the following false: $f(x)=2x+1$ produces a prime number if and only if x is prime
negative_passages.docid: 20980
negative_passages.text: Prove that $\frac{p^p-1}{p-1}$ is not prime if $p \equiv 1 \pmod 4$
negative_passages.docid: 102889
negative_passages.text: If $2^n - 1$ is prime from some integer $n$, prove that n must also be 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: 140412
negative_passages.text: If $p$ and $q$ are distinct primes such that $p|c$ and $q|c$, prove that $pq|c$
negative_passages.docid: 131551
negative_passages.text: $pq\mid pn^q+qn^p+kn$ for some $k,\,$ if $p,q$ are primes
negative_passages.docid: 6700
negative_passages.text: An Olympiad problem in prime numbers :can we proof that "n" is prime in this case "
negative_passages.docid: 83721
negative_passages.text: If a prime $p\mid ab$, then $p\mid a$ or $p\mid b\ $ [Euclid's Lemma]
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: 59036
negative_passages.text: If $a^{n}-1$ is prime then $a=2$ and $n$ is prime?
negative_passages.docid: 37947
negative_passages.text: Prove that if a number $n > 1$ is not prime, then it has a prime factor $\le \sqrt{n}$.
negative_passages.docid: 39915
negative_passages.text: Let $n$ be an integer such that if $d | n$ then $d + 1 | n + 1.$ Show that $n$ is a prime number
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: 14221
negative_passages.text: How can we prove it? ${a}{b}≤\frac{a^p}{p}+\frac{b^q}{q}$
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: 85287
negative_passages.text: Prove that, for $p > 3$, $(\frac 3p) = 1$ when $p \equiv 1,11 \pmod{12}$ and $(\frac 3p) = -1$ when $p \equiv 5,7 \pmod{12}$