query: Find all $n\in\mathbb{Z}^+$ such that the sum of the digits of $5^n$ equals $2^n$
positive_passages.docid: 143501
positive_passages.text: Find all $n\in\mathbb N^+$ such that the sum of the digits of $5^n$ is equal to $2^n$.
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 $2^{n}$.
negative_passages.docid: 5043
negative_passages.text: Show that $2^n>1+n\sqrt{2^{n-1}}$
negative_passages.docid: 90806
negative_passages.text: Find all positive integers $n$ such that $n$ is divisible by all the positive integers less than or equal to $\sqrt{n}$
negative_passages.docid: 148906
negative_passages.text: Finding integers $m,n$ such that $(m-n)^2=4mn/(m+n-1)$
negative_passages.docid: 1379
negative_passages.text: Summation of $n^2x^n$ terms
negative_passages.docid: 66568
negative_passages.text: Number Theory : Show that $\sum_{i=0}^\infty$ $ [\frac{n}{2^i}+\frac{1}{2}]$ $=$ $2n$
negative_passages.docid: 10589
negative_passages.text: How to factorize $5$ in $\mathbb{Z}[\root 3 \of 2]$?
negative_passages.docid: 82083
negative_passages.text: Solving $(z+1)^5 = z^5$
negative_passages.docid: 44577
negative_passages.text: Finding a formula to sum natural numbers up to $n$
negative_passages.docid: 17306
negative_passages.text: To find all $n$ such that $(n-1)!+1$ is a perfect power of $n$
negative_passages.docid: 74898
negative_passages.text: $1^n +2^n + \cdots +(p-1)^n \mod p =$?
negative_passages.docid: 25338
negative_passages.text: Sum of $n$ integers from set of size $2n-1$ divides $n$
negative_passages.docid: 54355
negative_passages.text: $1+x+\ldots+x^n$ perfect square
negative_passages.docid: 91113
negative_passages.text: $ \exists a, b \in \mathbb{Z} $ such that $ a^2 + b^2 = 5^k $
negative_passages.docid: 15268
negative_passages.text: Find all $x,y \in \mathbb N^*$ such that $\frac{1}{x} + \frac{1}{y}=5$
negative_passages.docid: 131708
negative_passages.text: Sum of powers of natural numbers