query: $1!+2!+3!+4!+5!+...+n!$ is a square of an integer
positive_passages.docid: 83862
positive_passages.text: How to prove that the number $1!+2!+3!+...+n!$ is never square?
negative_passages.docid: 59006
negative_passages.text: Proof of $\gcd(n!+1, (n+1)!+1)=1$?
negative_passages.docid: 22073
negative_passages.text: Prove by induction that $n^2<n!$
negative_passages.docid: 115244
negative_passages.text: Integer solutions of $m^2+(m+1)^2=n^4+(n+1)^4$
negative_passages.docid: 9920
negative_passages.text: $(n^2 - 1)!$ is a multiple of $(n!)^n$
negative_passages.docid: 65757
negative_passages.text: Prove that $n! + k$ is a composite number
negative_passages.docid: 142900
negative_passages.text: Find all positive integers a,b,c to the equation a!b!= a!+b!+c!
negative_passages.docid: 89031
negative_passages.text: Solve $a!b!=a!+b!+c!$ where $a$, $b$ and $c$ are nonnegative integers.
negative_passages.docid: 86935
negative_passages.text: Finding all the numbers that fit $x! + y! = z!$
negative_passages.docid: 157746
negative_passages.text: Finding all positive integer solutions to $(x!)(y!) = x!+y!+z!$
negative_passages.docid: 144836
negative_passages.text: How to prove that $\frac{(mn)!}{m!(n!)^m}$ is an integer?
negative_passages.docid: 126889
negative_passages.text: Proof by induction: inequality $n! > n^3$ for $n > 5$
negative_passages.docid: 72600
negative_passages.text: Proof that $n!\leq {(\frac{n+1}{2})}^{n}$
negative_passages.docid: 20691
negative_passages.text: prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer
negative_passages.docid: 116054
negative_passages.text: The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer
negative_passages.docid: 58589
negative_passages.text: On the factorial equations $A! B! =C!$ and $A!B!C! = D!$
negative_passages.docid: 63553
negative_passages.text: Show that $\frac{1}{(n+1)!}(1+\frac{1}{(n+1)}+\frac{1}{(n+1)^2}+\frac{1}{(n+1)^3}+\cdots)=\frac{1}{n!n}$