Show that there are infinitely many composite numbers \(n\) such that \(3^{n-1}-2^{n-1}\) is divisible by \(n\).
Show that there are infinitely many integers \(n\) such that \(2^n+1\) is divisible by \(n\). Find all prime numbers that satisfy this property.
If \(k>1\), show that \(k\) does not divide \(2^{k-1}+1\). Find all prime numbers \(p,q\) such that \(2^p+2^q\) is divisible by \(pq\).
Find all pairs \((x,n)\) of positive integers such that \(x^n + 2^n + 1\) is a divisor of \(x^{n+1} + 2^{n+1} + 1\).
Let \(n>1\) be an integer. Show that \(n\) does not divide \(2^n-1\).
Find all integers \(n\) such that \(1^n + 2^n + ... + (n-1)^n\) is divisible by \(n\).
How many integers are there \(n>1\) such that \(a^{25}-a\) is divisible by \(n\) for every integer \(a\).
Let \(p\) be a prime number, \(a\) be an integer, not divisible by \(p\). Prove that \(a^p-a\) is divisible by \(p\).
Let \(n\) be an integer. Denote by \(\phi(n)\) the number of integers from \(1\) to \(n-1\) coprime to \(n\). Find \(\phi(n)\) in the following cases:
\(n\) is a prime number.
\(n = p^k\) for a prime \(p\).
\(n=pq\) for two different primes \(p\) and \(q\).
Let \(n\) be an integer and let \(a\) be an integer coprime to \(n\). Prove that \(a^{\phi(n)-1}-1\) is divisible by \(n\).