I am looking at two Euler's theorems in my textbook which are the following:
If $p$ is prime and $a$ is any whole number, then $(a+1)^p - (a^p + 1) $ is evenly divisible by $p$.
If $p$ is prime and $a$ is any whole number, then p divides evenly into $a^p - a$.
I clearly understand Euler's proof for the first theorem and when I looked at the proof for the second theorem, I am little lost. Euler applies the first theorem in the proof of the second theorem. You can look at the proof of the second theorem here
And I am wondering whether $a^p-a$ can be factored in to $(a+1)^p - (a^p + 1) $. If so, please tell me how to factor $a^p-a$.
THANK YOU!