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You can also use Bertrand's postulate to show that no non-trivial factorial equals a perfect integer power greater than 1. An interesting question is whether y
by ykonstant 3y ago
You can also use Bertrand's postulate to show that no non-trivial factorial equals a perfect integer power greater than 1.
An interesting question is whether you can show this without resorting to Bertrand's postulate.
- perihelions 3y agoBecause {2..n} contains some prime p, in the range n/2 < p < n, which appears exactly once in the factorization of n!. (Is that the argument?)
- ykonstant 3y agoCorrect.
- contravariant 3y agoYou're going to need some kind of result on the occurence of primes at least, because if you remove all factors greater than some prime you trivially get perfect powers (e.g. 6! ≈ 12² if you ignore the factor 5). (After thinking about it some more, perhaps not trivially.)