CARMICHAEL CONJECTURE ON THE EULER FUNCTION IS VALID BELOW 10(10,000,000)

Carmichael's conjecture states that if phi(x) = n, then phi(y) = n for some y not-equal x (phi is Euler's totient function). We show that th e conjecture is valid for all x under 10(10,900,000). The main new ide a is the application of a prime-certification technique that allows us to very quickly certify the primality of the thousands of large numbe rs that must divide a counterexample.