State and prove Fermat's Little Theorem.
An odd number is called a Carmichael number if it is not prime, but every positive integer satisfies . Show that a Carmichael number cannot be divisible by the square of a prime. Show also that a product of two distinct odd primes cannot be a Carmichael number, and that a product of three distinct odd primes is a Carmichael number if and only if divides divides and divides . Deduce that 1729 is a Carmichael number.
[You may assume the result that, for any prime , there exists a number g prime to such that the congruence holds only when is a multiple of . The prime factors of 1729 are 7,13 and 19.]