In mathematics, Lehmer's totient problem asks whether there is any composite numbern such that Euler's totient function φ(n) divides n − 1. This is an unsolved problem.
It is known that φ(n) = n − 1 if and only if n is prime. So for every prime numbern, we have φ(n) = n − 1 and thus in particular φ(n) divides n − 1. D. H. Lehmer conjectured in 1932 that there are no composite numbers with this property.[1]
History
Lehmer showed that if any composite solution n exists, it must be odd, square-free, and divisible by at least seven distinct primes (i.e. ω(n) ≥ 7). Such a number must also be a Carmichael number.
In 1980, Cohen and Hagis proved that, for any solution n to the problem, n > 1020 and ω(n) ≥ 14.[2]
In 1988, Hagis showed that if 3 divides any solution n, then n > 101937042 and ω(n) ≥ 298848.[3] This was subsequently improved by Burcsi, Czirbusz, and Farkas, who showed that if 3 divides any solution n, then n > 10360000000 and ω(n) ≥ 40000000.[4]
A result from 2011 states that the number of solutions to the problem less than is at most .[5]
Cohen, Graeme L.; Hagis, Peter, jun. (1980). "On the number of prime factors of n if φ(n) divides n−1". Nieuw Arch. Wiskd. III Series. 28: 177–185. ISSN0028-9825. Zbl0436.10002.{{cite journal}}: CS1 maint: multiple names: authors list (link)