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Carmichael’s conjecture on the Euler function is valid below \(10^{10,000,000}\). (English) Zbl 0801.11001

Summary: Carmichael’s conjecture states that if \(\varphi(x) =n\), then \(\varphi(y) =n\) for some \(y\neq x\) (\(\varphi\) 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.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11Y70 Values of arithmetic functions; tables
11A41 Primes
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