Schlafly, Aaron; Wagon, Stan Carmichael’s conjecture on the Euler function is valid below \(10^{10,000,000}\). (English) Zbl 0801.11001 Math. Comput. 63, No. 207, 415-419 (1994). 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. Cited in 1 ReviewCited in 2 Documents MSC: 11A25 Arithmetic functions; related numbers; inversion formulas 11Y70 Values of arithmetic functions; tables 11A41 Primes Keywords:Carmichael’s conjecture; Euler’s totient function; prime-certification PDFBibTeX XMLCite \textit{A. Schlafly} and \textit{S. Wagon}, Math. Comput. 63, No. 207, 415--419 (1994; Zbl 0801.11001) Full Text: DOI