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

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

  
but is there other solutions...
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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)
by
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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.
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Unfortunately this is one of the hardest types of problems: hard to find another solution and hard to prove there are no more solutions.
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I got (4,4),(19,29),(20,29),(22,29),(24,29) and many others by running a program

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