query: Find all positive integers satisfying $\frac{2^n+1}{n^2} =k $
positive_passages.docid: 19254
positive_passages.text: How many rationals of the form $\large \frac{2^n+1}{n^2}$ are integers?
negative_passages.docid: 55100
negative_passages.text: The positive integer solutions for $2^a+3^b=5^c$
negative_passages.docid: 149489
negative_passages.text: 30 positive integers
negative_passages.docid: 46668
negative_passages.text: How to find integer solutions to $M^2=5N^2+2N+1$?
negative_passages.docid: 103555
negative_passages.text: Find all positive integral soludions of $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4$
negative_passages.docid: 115244
negative_passages.text: Integer solutions of $m^2+(m+1)^2=n^4+(n+1)^4$
negative_passages.docid: 326
negative_passages.text: Find $\sum_{k=1}^{\infty}\frac{1}{2^{k+1}-1}$
negative_passages.docid: 20834
negative_passages.text: Integer solutions to $c=a^2-b^2$
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: 134834
negative_passages.text: $n^2 + (n+1)^2 = m^3$ has no solution in the positive integers
negative_passages.docid: 33810
negative_passages.text: Proving there are an infinite number of pairs of positive integers $(m,n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer
negative_passages.docid: 45090
negative_passages.text: Given any positive real numbers $a,b,c$, we have $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$
negative_passages.docid: 50927
negative_passages.text: If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$
negative_passages.docid: 8884
negative_passages.text: Find all natural numbers $n$ and $m$ for which $n^m=(n-1)!+1$
negative_passages.docid: 124241
negative_passages.text: Finding all integers $k \geq 2$ such that $k^2 \equiv 5k \pmod{15}$. What is going on here?
negative_passages.docid: 146439
negative_passages.text: Is every sufficiently large positive integer of the form $ab + ac + bc + 1$?
negative_passages.docid: 155363
negative_passages.text: Find all integers $(x, y)$ such that $1 + 2^x + 2^{2x + 1} = y^2$