Problema do Rousseau

["Luis Lopes" <llopes@xxxxxxxxxxxxx>]

Sauda,c~oes,

O prof. Rousseau me prop^os o seguinte problema:

The following problem appears in the most recent Pi Mu Epsilon Journal, 
and so far I haven't been able to solve it. 

The Smarnandache function S is defined as follows:
S(n) is the smallest integer m such that n divides m!.  Prove that there
is always a prime between S(n) and S(n+1)  (including the end values).

The first few values are S(1) = 0, S(2)=2, S(3)=3, S(4)=4, S(5)=5,
S(6)=3, S(7)=7, S(8)=4, S(9)=6, S(10)=5.  In general S(n) = max S(p^k)
where the maximum is taken over all factors p_1^{k_1}, p_2^{k_2},
\ldots, in the canonical prime factorization of n.  Of course, if either
S(n) or S(n+1) is prime, there is nothing to prove.  Thus the first
"interesting" case above is S(8)=4, S(9)=6, where p=5 is the prime
between S(8) and S(9).

[ ]'s
Lu'is