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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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

Authors: Aaron Schlafly and Stan Wagon
Journal: Math. Comp. 63 (1994), 415-419
MSC: Primary 11A25; Secondary 11A51, 11Y11
MathSciNet review: 1226815
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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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Keywords: Euler’s function, Carmichael’s conjecture, prime certification
Article copyright: © Copyright 1994 American Mathematical Society