- 4

- 0

What would be the converse of this conjecture, How can you prove whether or not the converse holds?So you are asking, "Is it true that n^{p}- n is divisible by p for p any prime?"

Yes, it is and for exactly the reason you state: if p is prime then [itex]\left(\begin{array}{c}p \\ i\end{array}\right)[/itex] for p prime and 0< i< p is divisible by p. That itself can be proven directly from the definition:

[tex]\left(\begin{array}{c}p \\ i\end{array}\right)= \frac{p!}{i!(p-i)!}[/tex]

as long as i is neither 0 nor p, 0<p-i< p and so neither i! nor (p- i)! have a factor of p. Since p! does, the binomial coefficient is divisible by p. (We need p to be prime so that other factors in i! and (p- i)! do not "combine" to cancel p.)

Now, to show that n^{p}- n is divisible by p, do exactly what mathman suggested.

First, when n= 1, 1^{p}- 1= 0 which is divisible by p. Now assume the statement is true for some k: k^{p}- k= mp for some integer m. Then, by the binomial theorem,

[tex](k+1)^p= \sum_{i=0}^p \left(\begin{array}{c}p \\ i\end{array}\right) k^i[/itex]

subtracting k+1 from that does two things: first it cancels the i=0 term which is 1. Also we can combine the "k" with the i= p term which is k^{p}so we have k^{p}- k= mp. The other terms, all with 0< i< p, contain, as above, factors of p.

thanks