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 Infinite Illation Decision 2 (Posted on 2016-01-15)
Does there exist an infinite number of values of a positive integers N such that N! is divisible by N2 + 1?

 No Solution Yet Submitted by K Sengupta No Rating

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 Preliminary search and the OEIS sequence Comment 1 of 1
Clearly N^2+1 must be composite.  Furthermore, all the prime factors of N^2+1 are at most N.  This is a necessary but not sufficient requirement (N=7 has N^2+1=50=2*5*5 but 50 does not divide 7!=5040).

After manually trying N up to 40, I found 18, 21, and 38 work.  This was enough to find OEIS A120416, which has the first 36 such numbers.

I don't see any pattern in the sequence but values of N seem to be plentiful with the largest jump being 17, going from 21 to 38.

 Posted by Brian Smith on 2016-01-15 13:01:56

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