query: How can I prove that if $2^x+1$ is prime, then $x$ is a power of $2$?
positive_passages.docid: 2802
positive_passages.text: If $2^n+1$ is prime, why must $n$ be a power of $2$?
negative_passages.docid: 22742
negative_passages.text: Prove that if $p$ and $p^2+2$ are prime then $p^3+2$ is prime too
negative_passages.docid: 5043
negative_passages.text: Show that $2^n>1+n\sqrt{2^{n-1}}$
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: 70308
negative_passages.text: Prove that if $p$ and $p+2$ are both prime integers greater than $3$, then $6$ is a factor of $p+1$.
negative_passages.docid: 30374
negative_passages.text: If $n\in\mathbb N$ and $4^n+2^n+1$ is prime, prove that there exists an $m\in\mathbb N\cup\{0\}$ such that $n=3^m$.
negative_passages.docid: 157884
negative_passages.text: For any prime $p>3$ show that 3 divides $2p^2+1$
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: 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: 44403
negative_passages.text: Prove that $|a+b|^p \leq 2^p \{ |a|^p +|b|^p \}$
negative_passages.docid: 21629
negative_passages.text: Using Fermat's Little Theorem Prove if $p$ is prime, prove $1^p + 2^p + 3^p +...+(p-1)^p \equiv 0 \bmod{p}$
negative_passages.docid: 34666
negative_passages.text: Prove that the expression cannot be a power of 2
negative_passages.docid: 153294
negative_passages.text: What is the infinite product of (primes^2+1)/(primes^2-1)?
negative_passages.docid: 119657
negative_passages.text: If $ f _p(x)=1+x+x^2+ \cdots +x^{p-1} $ where $p$ is prime number, $e$ is any positive integer.
negative_passages.docid: 153862
negative_passages.text: A question about a primitive root mod $p=2^{2^k}+1$, where $p$ is prime.
negative_passages.docid: 34588
negative_passages.text: Number Theory: $p!+p=n^2$, $p$ = prime
negative_passages.docid: 34954
negative_passages.text: If $11..11$($n$ times 1) is prime, prove that $n$ is also prime