query: If $p$ is prime, then for any integer $m$, $0 < m < p$, there is a unique $n$, $0 < n < p$, such that $mn$ is congruent to $1\pmod p$
positive_passages.docid: 2904
positive_passages.text: Solving linear congruence (modular inverse or fraction) via gcd Bezout equation
negative_passages.docid: 142584
negative_passages.text: Confusion in proof that primes $p = 4k + 1$ are uniquely the sums of two squares
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: 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: 103732
negative_passages.text: Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.
negative_passages.docid: 93898
negative_passages.text: Prove that for any prime p, there is a complete set of mod p residues {q1, . . . , qp}, where each qi is prime.
negative_passages.docid: 18634
negative_passages.text: Combinations of sets raised to the power of a prime modulus: $\binom{p^{\alpha}-1}k \equiv (-1)^k \pmod p$
negative_passages.docid: 154305
negative_passages.text: Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$
negative_passages.docid: 5841
negative_passages.text: Proving prime $p$ divides $\binom{p}{k}$ for $k\in\{1,\ldots,p-1\}$
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: 66851
negative_passages.text: If $p,q$ are primes $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$
negative_passages.docid: 97021
negative_passages.text: Do there exist two primes $p<q$ such that $p^n-1\mid q^n-1$ for infinitely many $n$?
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: 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: 75123
negative_passages.text: If $p$ be a prime and r be any integer, $0 < r < p$ then $\frac{(p-1)!}{r!(p-r)!}$ is an integer.
negative_passages.docid: 23153
negative_passages.text: $\bmod p\!:\ x^2\equiv 1\iff x\equiv \pm1\ $ for prime $p$