query: prove that the number $38^n+31$ is composite
positive_passages.docid: 152008
positive_passages.text: Least prime of the form $38^n+31$
negative_passages.docid: 124240
negative_passages.text: Inductive proof of $\,9 \mathrel| 4^n+6n-1\,$ for all $\,n\in\Bbb N$
negative_passages.docid: 26449
negative_passages.text: $|G|=24$ prove that $G$ is not simple
negative_passages.docid: 151928
negative_passages.text: How to prove that $N\setminus A$ is finite?
negative_passages.docid: 109599
negative_passages.text: A finite sum involving the binomial coefficients and the harmonic numbers
negative_passages.docid: 1895
negative_passages.text: How to show that $2730\mid n^{13}-n\;\;\forall n\in\mathbb{N}$
negative_passages.docid: 22367
negative_passages.text: How do I prove that for every positive integer $n$, there exist $n$ consecutive positive integers, each of which is composite?
negative_passages.docid: 46400
negative_passages.text: how do i prove that $17^n-12^n-24^n+19^n \equiv 0 \pmod{35}$
negative_passages.docid: 103504
negative_passages.text: Weird $3^n$ in an identity to be combinatorially proved
negative_passages.docid: 44705
negative_passages.text: Prove that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$
negative_passages.docid: 65757
negative_passages.text: Prove that $n! + k$ is a composite number
negative_passages.docid: 24444
negative_passages.text: Prove that $371\cdots 1$ is not prime
negative_passages.docid: 56574
negative_passages.text: For any integer $a,b$ let $N_{a,b}$ denote the number of positive integer $x<1000$ satisfying $x= a( mod\;27)$ and $x=b(mod\;37)$. Then
negative_passages.docid: 96027
negative_passages.text: Prove that if $a+b+c$ divides $abc$, then $a+b+c$ must be composite.
negative_passages.docid: 132526
negative_passages.text: I'm trying to find the longest consecutive set of composite numbers
negative_passages.docid: 123155
negative_passages.text: If $mn+1$ is a multiple of $24$, prove that $m+n$ is also a multiple of 24.
negative_passages.docid: 43033
negative_passages.text: Proof on Catalan numbers