Question

For every positive integer a , find a composite number n such that n|a^n-a.?

Answer 1

Answer:

Step-by-step explanation:This solution is from "Elementary Number Theory" by Sierpinski:

n|an−a

If a is composite we may set n=a then n|an−a. If a=1 we may take for example n=4 because 14−1 is divisible by 4. When a is a prime greater than 2 we can take n=2a because in this case a is odd and even number a2n−a is divisible by n and 2 and consequently by 2a=n.

It remains only the case where a=2. In this case we may let n=341=11×31, which gives 341|2341−2 because:

210−1≡0mod11

⇒2340−1≡0mod11

which means 2341−2≡0mod11

also:

25−1≡2340−1≡0mod31

therefore :

11×31=341|2341−2