query: For how many numbers between 1 and 50 will $\frac{(n^2 - 1)!}{n!^n}$ be an integer?
positive_passages.docid: 9920
positive_passages.text: $(n^2 - 1)!$ is a multiple of $(n!)^n$
negative_passages.docid: 23606
negative_passages.text: Prime divisors of the integer $n^2+n-1$ (using the Legendre symbol)
negative_passages.docid: 119278
negative_passages.text: The number $\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^n\right]$ is always an integer
negative_passages.docid: 148647
negative_passages.text: If $\dfrac{4x^2-1}{4x^2-y^2}$ is an integer, then it is $1$
negative_passages.docid: 52241
negative_passages.text: Calculate sum of series $\sum \frac{n^2}{n!}$
negative_passages.docid: 31866
negative_passages.text: Calculating the limit $\lim((n!)^{1/n})$
negative_passages.docid: 148803
negative_passages.text: 501 distinct coprime integers between 1 and 1000?
negative_passages.docid: 63661
negative_passages.text: integers $n$ such that $(n-1)!+1$ is a power of $n$
negative_passages.docid: 110383
negative_passages.text: How many numbers less than $m$ and relatively prime to $n$, where $m>n$?
negative_passages.docid: 151153
negative_passages.text: Prove $\frac{a(a^2+2)}{3}$ is an integer for all integer $a\geqslant 1$
negative_passages.docid: 152870
negative_passages.text: Near integers in powers of binomials with radicals
negative_passages.docid: 29609
negative_passages.text: Limits involving factorials $\lim_{N\to\infty} \frac{N!}{(N-k)!N^{k}}$
negative_passages.docid: 118383
negative_passages.text: If $n = 51! +1$, then find number of primes among $n+1,n+2,\ldots, n+50$
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: 125538
negative_passages.text: limit of $n^{\frac{1}{\sqrt{n}}}$
negative_passages.docid: 4024
negative_passages.text: Finding the limit of $\frac {n}{\sqrt[n]{n!}}$
negative_passages.docid: 144836
negative_passages.text: How to prove that $\frac{(mn)!}{m!(n!)^m}$ is an integer?