query: Suppose we fix $n$, does there always exist a prime $p$ such that $p\equiv 1$ modulo $n$?
positive_passages.docid: 52530
positive_passages.text: There exists a prime congruent to $m$ mod $p$
negative_passages.docid: 151985
negative_passages.text: If $p$ is a prime and $x,y \in \mathbb{Z}$, then $(x+y)^p \equiv x^p+y^p \pmod{p}$
negative_passages.docid: 10266
negative_passages.text: Minimal polynomial reducible modulo every prime $p$
negative_passages.docid: 18108
negative_passages.text: Show if $(a,p)=1$ there is a unique inverse of $a$ modulo $p$
negative_passages.docid: 5142
negative_passages.text: Infinitely many primes of the form $pn+1$
negative_passages.docid: 8710
negative_passages.text: Bounds for $n$-th prime
negative_passages.docid: 71788
negative_passages.text: Prove existence of multiplicative inverse modulo p when p 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: 119051
negative_passages.text: Show that only finitely many positive integers $n$ have the property that for all $m$ for which $1<m<n$ and $\text{gcd}(m,n) = 1$, $m$ is prime.
negative_passages.docid: 50781
negative_passages.text: $(n+1)$th prime $p_{n+1}$ less than or equal to $p_1p_2\dots p_n+1$
negative_passages.docid: 62996
negative_passages.text: A problem regarding the proof of ${p^nk\choose p^n}\equiv k\mod p$, where $p\nmid k$.
negative_passages.docid: 65826
negative_passages.text: Find a prime number $p$ and an integer $b<p$ such that $p$ divides $b^{p−1}−1$.
negative_passages.docid: 59442
negative_passages.text: Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?
negative_passages.docid: 94917
negative_passages.text: Prove that for every prime $p$, there exists a solution to $(x^2-2)(x^2-6)(x^2-3) \equiv 0 \mod(p)$
negative_passages.docid: 122612
negative_passages.text: An integer is a $n$'th power if that holds true for all moduli.
negative_passages.docid: 2218
negative_passages.text: Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime [Wilson's Theorem]
negative_passages.docid: 130342
negative_passages.text: Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.