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Lehmer's totient problem

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Unsolved problem in mathematics:
Can the totient function of a composite number divide ?

In mathematics, Lehmer's totient problem asks whether there is any composite number n 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 number n, 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

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  • 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]

References

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  1. ^ Lehmer (1932)
  2. ^ Sándor et al (2006) p.23
  3. ^ Guy (2004) p.142
  4. ^ Burcsi, P., Czirbusz, S., Farkas, G. (2011). "Computational investigation of Lehmer's totient problem". Ann. Univ. Sci. Budapest. Sect. Comput. 35: 43–49.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  5. ^ Luca and Pomerance (2011)