# does n!m! = t! have infinitely many solutions? besides trivial ones

Does n!m!=t! have infinitely many solutions in positive interger besides trivial ones? (n=0 m=1 etc)

Can't work this one out. thanks.

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$(n!)!=(n!-1)!\cdot n!$ - is it trivial or not? –  Nurdin Takenov Sep 20 '10 at 11:37
I'm not sure if this is considered trivial or not but $n! = (n!)!/(n!-1)!$. Let $n < m$, since $n! = t!/m!$ the range $(m,t]$ must avoid primes because if there was a prime in there $n!$ wouldn't contain it. –  muad Sep 20 '10 at 11:38
Also - there is related question - mathoverflow.net/questions/39210/… Because, if there are any $(m,n)$, such that $n!=m(m+1)$, then we have:$(m+1)!=(m-1)!\cdot n!$ –  Nurdin Takenov Sep 20 '10 at 12:11
The OEIS sequence oeis.org/classic/A003135 is related to this question. –  tdnoe Sep 20 '10 at 18:21
Kevin, are you thinking of Guy's book, Unsolved Problems In Number Theory? The reference is problem B23. –  Gerry Myerson Sep 21 '10 at 4:38

$(n!)!=(n!-1)!\cdot n!$ - is it trivial or not?