Lehmer's totient problem
Euler's totient counts the integers in that are coprime to . When is prime, , which certainly divides . Is there a composite number for which divides ?
Posed by D. H. Lehmer, 1932
History and references
Still open as of 2026.
Derrick Henry Lehmer asked the question in 1932. He showed that a composite solution would have to be odd, squarefree and divisible by at least seven distinct primes, and such a number would also be a Carmichael number. In 1980 Cohen and Hagis showed that any solution must exceed and have at least 14 prime factors, and later work pushed these bounds much further in special cases. Luca and Pomerance proved in 2011 that solutions, if they exist, are very sparse. No solution is known and the question is open.
References
Solutions 0
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