query: Find all positive integers $n$ for which $(n-1)!+1$ is a power of $n$
positive_passages.docid: 17306
positive_passages.text: To find all $n$ such that $(n-1)!+1$ is a perfect power of $n$
negative_passages.docid: 8034
negative_passages.text: Can $n!$ be a perfect square when $n$ is an integer greater than $1$?
negative_passages.docid: 115258
negative_passages.text: Find all positive integers $n$ such that $\phi(n)+\sigma(n)=2n$.
negative_passages.docid: 88144
negative_passages.text: Induction Proof $n! < n^n$
negative_passages.docid: 153763
negative_passages.text: Prove that $a_1!\cdots a_k! < n!$ whenever $a_1+\cdots+a_k < n$
negative_passages.docid: 8609
negative_passages.text: Prove by induction that $n!>2^n$
negative_passages.docid: 100558
negative_passages.text: Prove that for any even positive integer $n$, $n^2-1 \mid 2^{n!}-1$
negative_passages.docid: 118039
negative_passages.text: How prove $p!|(a+1)$ if $p!|a^p+1$
negative_passages.docid: 106888
negative_passages.text: Factorial identity $n!=1+(1-1/1!)n+(1-1/1!+1/2!)n(n-1)+\cdots$
negative_passages.docid: 5206
negative_passages.text: Highest power of a prime $p$ dividing $N!$
negative_passages.docid: 43865
negative_passages.text: Find all positive integers $n$ such that $\phi(n)=6$.
negative_passages.docid: 99453
negative_passages.text: Sum of $k$-th powers
negative_passages.docid: 142008
negative_passages.text: Find all natural numbers $n$ such that ${(n+1)}^{1/2}-{(n-1)}^{1/2}$ is rational
negative_passages.docid: 16235
negative_passages.text: Prove by induction that every positive integer is represented by some binary number?
negative_passages.docid: 152870
negative_passages.text: Near integers in powers of binomials with radicals
negative_passages.docid: 52542
negative_passages.text: For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$?
negative_passages.docid: 8504
negative_passages.text: Factoring $x^n + y^n$ over the integers