(In reply to computer exploration (spoiler? -- UBASIC program)
If the (M-3) consecutive integers are from 5 to M+1, then M! must equal M!*(M+1)/24, and that is indeed the M=23 that was found.
If the integers were from 6 to M+2, then M! must equal M!*(M+1)*(M+2)/120, so (M+1)*(M+2) = 120, which doesn't have a positive integral solution.
From 7 to M+3: (M+1)*(M+2)*(M+3)= 720, which is satisfied by M = 7
These necessarily have smaller solutions in M as we continue this process, as the denominator of the fraction that is to equal 1 goes up by a smaller factor than the numerator when M is large.
Posted by Charlie
on 2008-10-24 13:11:27