Question

Find all positive integers m, n such that (m!+n!+mn) is a perfect square

Answer 1

I found that m=4 and n=4 is a solution ...
  

  
but is there other solutions...

Answer 2

There's a few solutions with n=0 (if you want to count these)

    0! + 0! + 0·0  =  1²
    1! + 0! + 1·0  =  1²
    4! + 0! + 4·0  =  5²
    5! + 0! + 5·0  =  11²
    7! + 0! + 7·0  =  71²
    (Possibly more)

Answer 3

Can Wilson theorem utilize for it? Becouse all n^(2)=0 or 1 (mod 3)... We need to determinet m is prime or not. Becouse system is simmetric so no matter what we chose for test.

Answer 4

Unfortunately this is one of the hardest types of problems: hard to find another solution and hard to prove there are no more solutions.

Answer 5

I got (4,4),(19,29),(20,29),(22,29),(24,29) and many others by running a program