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

Schlafly, Aaron; Wagon, Stan

doi:10.1090/S0025-5718-1994-1226815-3

Carmichael’s conjecture on the Euler function is valid below $10^ {10,000,000}$
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by Aaron Schlafly and Stan Wagon PDF

Math. Comp. 63 (1994), 415-419 Request permission

Abstract:

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

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