Carmichael's Conjecture on the Euler Function is Valid Below 10^{10,000, 000}

Abstract

CarmichaeFs conjecture states that if ix) = n , then (y) = n for some y ^ x (^ is Euler's totient function). We show that the conjecture is valid for all x under io10'900'000 . The main new idea is the application of a prime-certification technique that allows us to very quickly certify the primality of the thousands of large numbers that must divide a counterexample.