Find three different positive integers whose factorials are each one less than a perfect square, such that the total of all three factorials is also a perfect square.
The required three different positive integers are 4, 5 and 7
4! = 24 = 5^2-1
5! = 120 = 11^2 -1
7! = 5040 = 71^2 -1
and, 4!+5!+7! = 24+120+5040=5184= 72^2
This is also mentioned in Sloane's Online Encyclopedia of Integer sequences as A146968: Brocard's problem: positive integers n such that n!+1 = m^2 in the following location:
http://oeis.org/A085692
Edited on January 5, 2023, 2:06 am
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