The statement of this problem is elementary... How about the proof ? Let $p$ be an odd prime number. For every integer $a$ , define the number

$ S_{a}=\frac{a}{1}+\frac{a^{2}}{2}+....+\frac{a^{p-1}}{p-1} $.

Let $m$ and $n$ be integers such that $ S_{3}+S_{4}-3S_{2}=\frac{m}{n} $. Prove that $p$ divides $m$.

QED. – Noam D. Elkies Apr 29 '12 at 16:07lookslike a logarithm...) – Noam D. Elkies Apr 29 '12 at 22:57