query: Proving that $7n+3$ is prime infinitely often?
positive_passages.docid: 10596
positive_passages.text: Dirichlet's theorem on primes in arithmetic progression
negative_passages.docid: 23544
negative_passages.text: Complete induction proof that every $n > 1$ can be written as a product of primes
negative_passages.docid: 24053
negative_passages.text: How can I prove that $n^7 - n$ is divisible by $42$ for any integer $n$?
negative_passages.docid: 144592
negative_passages.text: how to prove the existence and uniqueness of the prime factorization of any natural number.
negative_passages.docid: 15915
negative_passages.text: For any prime $p > 3$, why is $p^2-1$ always divisible by 24?
negative_passages.docid: 75581
negative_passages.text: Is the sum of the first $n$ primes a prime infinitely many times?
negative_passages.docid: 65131
negative_passages.text: Can infinitely many primes lie over a prime?
negative_passages.docid: 157884
negative_passages.text: For any prime $p>3$ show that 3 divides $2p^2+1$
negative_passages.docid: 119486
negative_passages.text: Upper and Lower bounds on the nth prime number
negative_passages.docid: 90102
negative_passages.text: How to prove that $p-1$ is squarefree infinitely often?
negative_passages.docid: 111967
negative_passages.text: Let $p$ be a prime number. Prove that there exists $x∈Z$ for which $p|x^2−x+ 3$ if and only if there exists $y∈Z$ for which $p|y^2−y+ 25$.
negative_passages.docid: 135892
negative_passages.text: Understanding the proof for showing that there are infinitely many $n\in\mathbb{Z}^+$ such that $n!+1$ is divisible by at least two distinct primes
negative_passages.docid: 68221
negative_passages.text: Question regardles primes and the fundamental theorem of arithmetic
negative_passages.docid: 83935
negative_passages.text: Using congruences, show $\frac{1}{5}n^5 + \frac{1}{3}n^3 + \frac{7}{15}n$ is integer for every $n$
negative_passages.docid: 99428
negative_passages.text: Prove that $3^{2n-1} + 2^{n+1}$ is always a multiple of $7$.
negative_passages.docid: 45793
negative_passages.text: How do you prove a number is prime?
negative_passages.docid: 1189
negative_passages.text: Using Fermats prime numbers to prove that there is infinitely many prime numbers