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Carmichael’s conjecture states that if      ϕ  ( x ) = n \phi (x) = n      , then      ϕ  ( y ) = n \phi (y) = n       for some      y ≠  x y \ne x       (     ϕ  \phi       is Euler’s totient function). We show that the conjecture is valid for all  x  under      10 10 , 900 , 000 {10^{10,900,000}}      . 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.

Carmichael’s conjecture on the Euler function is valid below $10\sp {10,000,000}$