Alexander R. Povolotsky's problem 1

Posted in sci.math.research, NUMTHRY and primepuzzles.net. Referenced in OEIS's A145355. http://www.research.att.com/~njas/sequences/A145355OEIS is managed/edited by renowned scientist - Neil Sloane,therefore OEIS reference is a Reliable Source !So far proven only for the case, when k=2. Complements Brocard's conjecture.

Variants of the problem:

Dabrowski (1996) has shown that it would follow from the abc conjecture that

n! + A = k2

has only finitely many solutions, for any given integer A.The problem is communicated to Richard Guy for inclusion into next edition of his "Unsolved Problems" book.