query: If $2^n+1$ is prime then $n=2^k$ for some positive integer $k$.
positive_passages.docid: 42885
positive_passages.text: Fermat primes relation to $2^n+1$
negative_passages.docid: 50413
negative_passages.text: Proof: if $p$ is prime, and $0<k<p$ then $p$ divides $\binom pk$
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: 122612
negative_passages.text: An integer is a $n$'th power if that holds true for all moduli.
negative_passages.docid: 37947
negative_passages.text: Prove that if a number $n > 1$ is not prime, then it has a prime factor $\le \sqrt{n}$.
negative_passages.docid: 100984
negative_passages.text: Prove that if prime $p$ divide $a_{2k}-2$, then $p$ divide also $a_{2k+1}-1$.
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: 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: 63356
negative_passages.text: Can $x^{n}-1$ be prime if $x$ is not a power of $2$ and $n$ is odd?
negative_passages.docid: 77806
negative_passages.text: Proving if $a$ is not a multiple of prime $p$, then an integer $b$ exists such that $p^b - 1$ is a multiple of $a$
negative_passages.docid: 152299
negative_passages.text: Proving $a^{n-1} \equiv 1 \mod n $ when n is not prime.
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: 1837
negative_passages.text: Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$
negative_passages.docid: 158563
negative_passages.text: If $m^4+4^n$ is prime, then $m=n=1$ or $m$ is odd and $n$ even
negative_passages.docid: 12967
negative_passages.text: Find all $n>1$ such that $\dfrac{2^n+1}{n^2}$ is an integer.
negative_passages.docid: 107502
negative_passages.text: Let $p \geq 2$. Prove that if $2^p-1$ is prime then $p$ must also be prime.
negative_passages.docid: 39670
negative_passages.text: Use Pigeonhole to show of any set of $2^{n+1}-1$ positive integers is possible choose $2^{n}$ elements such that their sum is divisible by $2^{n}$.