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Find all positive integers m, n such that (m!+n!+mn) is a perfect square
playboy_col
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Jun 21, 2022
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Find all positive integers m, n such that (m!+n!+mn) is a perfect square
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playboy_col
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Jun 21, 2022
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I found that m=4 and n=4 is a solution ...
but is there other solutions...
FaireBien
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Jun 21, 2022
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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)
Bwb_2
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Jun 21, 2022
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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.
SFBARTable
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Jun 21, 2022
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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.
OpenAPTV
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Jun 21, 2022
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I got (4,4),(19,29),(20,29),(22,29),(24,29) and many others by running a program
abdullaalamre
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Jun 21, 2022
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abdullaalamre
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