# Why is $Z^{p}-Z$ is always a multiple of $p$, Where $p$ is a prime, and $Z$ is a positive integer?

I came upon the conclusion that $Z^{p}-Z$ is always a multiple of $p$. $\$Where $p$ is a prime, and $Z$ is a positive integer.

I have the following four questions:
a) Could someone provide a complete proof for the above statement?
b) If any counter examples can be found?
c) If this can be easily derived from properties of Mersenne primes in the case $Z=2$?
d) If this indeed true, does it already have a name, wikipedia page, etc... (I have failed to find mention of this fact (if it is true), but I would like to be directed to where I may learn more, if it is well known)

I have tested this with approxamitely 80 sets of numbers and it seems to hold up.

I have found a partial, and flawed proof, but it explains some cases. I originally started thinking about this problem when I realized that because 7 may be expressed as a repeating decimal with six repeating digits, the following must be true: $$\frac 1 7 * \frac n n = \frac n {{10}^6-1}$$ (because $\overline {0.000001} = \frac 1 {{10}^6-1}$, in this case $n = 142857$). $\$Hence, ${{10}^6-1}$ is a multiple of 7. $\$Since $10*({{10}^6-1})$ is still a multiple of seven, ${{10}^7-10}$ must clearly be a multiple of $7$, conforming with the original formula. I first tested the previously mentioned equation on the assumption that these steps should hold true in every base. $\$However, this would seem to apply only to numbers with $7$'s property of repeating every $n-1$ digits, and it would have to indeed hold true across bases. I hope a fuller expanation may be provided.

• In case you were unaware: en.wikipedia.org/wiki/Fermat%27s_little_theorem – Cameron Williams Mar 3 '16 at 3:52
• In fact, if $p$ is prime, then $p\mid z^p-z$ for any integer $z$ (not only positive integer). – user236182 Mar 3 '16 at 3:59
• There is a connection with Mersenne primes, since Fermat found the result while trying to narrow the possibilities for divisors of $2^p-1$. – André Nicolas Mar 3 '16 at 4:07