query: can $a_{n+1}=2a_n+1$ generate prime numbers?
positive_passages.docid: 151701
positive_passages.text: Does there exist an $a_0$ such that the sequence $a_{n+1} = 2a_n + 1$ is prime for all $n \ge 0$?
negative_passages.docid: 78342
negative_passages.text: When is $\phi(n)$ a prime number?
negative_passages.docid: 92737
negative_passages.text: primes whose difference is a multiple of $n$
negative_passages.docid: 50596
negative_passages.text: $p_1p_2...p_n +1$ is not square for $p_k = k$'th prime
negative_passages.docid: 130342
negative_passages.text: Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
negative_passages.docid: 13739
negative_passages.text: For all $n>2$ there exists a prime number between $n$ and $ n!$
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: 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: 64782
negative_passages.text: How to prove if $n$ is prime and $n | a^2$ then $n | a$?
negative_passages.docid: 152299
negative_passages.text: Proving $a^{n-1} \equiv 1 \mod n $ when n is not prime.
negative_passages.docid: 119486
negative_passages.text: Upper and Lower bounds on the nth prime number
negative_passages.docid: 154475
negative_passages.text: Prove that if $2^n - 1$ is prime, then $n$ is prime for $n$ being a natural number
negative_passages.docid: 81250
negative_passages.text: Infinitude of prime numbers
negative_passages.docid: 75581
negative_passages.text: Is the sum of the first $n$ primes a prime infinitely many times?
negative_passages.docid: 82482
negative_passages.text: Proving an infinite number of primes of the form 6n+1
negative_passages.docid: 143772
negative_passages.text: How can I algebraically prove that $2^n - 1$ is not always prime?
negative_passages.docid: 2802
negative_passages.text: If $2^n+1$ is prime, why must $n$ be a power of $2$?