Lehmer's totient problem

For Lehmer's Mahler measure problem, see Lehmer's conjecture.

Unsolved problem in mathematics:

Can the totient function of a composite number n divide n − 1?
(more unsolved problems in mathematics)

In mathematics, Lehmer's totient problem, named for D. H. Lehmer, asks whether there is any composite number n such that Euler's totient function φ(n) divides n − 1. This is true of every prime number, and Lehmer conjectured in 1932 that there are no composite solutions: he showed that if any such n exists, it must be odd, square-free, and divisible by at least seven primes (i.e. ω(n) ≥ 7). Such a number must also be a Carmichael number.

Properties

In 1980 Cohen and Hagis proved that, for any solution n to the problem, n > 10²⁰ and ω(n) ≥ 14.^[1]
In 1988 Hagis showed that if 3 divides any solution n then n > 10^1937042 and ω(n) ≥ 298848.^[2]
The number of solutions to the problem less than X is $O\left({X^{{1/2}}(\log X)^{{3/4}}}\right)$ .^[3]

References

↑ Sándor et al (2006) p.23
↑ Guy (2004) p.142
↑ Sándor et al (2006) p.24

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. Ser. 28: 177–185. ISSN 0028-9825. Zbl 0436.10002.
Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. B37. ISBN 0-387-20860-7. Zbl 1058.11001.
Hagis, Peter, jun. (1988). "On the equation M⋅φ(n)=n−1". Nieuw Arch. Wiskd., IV. Ser. 6 (3): 255–261. ISSN 0028-9825. Zbl 0668.10006.
Lehmer, D. H. (1932). "On Euler's totient function". Bulletin of the American Mathematical Society. 38: 745–751. doi:10.1090/s0002-9904-1932-05521-5. ISSN 0002-9904. Zbl 0005.34302.
Ribenboim, Paulo (1996). The New Book of Prime Number Records (3rd ed.). New York: Springer-Verlag. ISBN 0-387-94457-5. Zbl 0856.11001.
Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. ISBN 1-4020-4215-9. Zbl 1151.11300.
Burcsi, Péter; Czirbusz, Sándor; Farkas, Gábor (2011). "Computational investigation of Lehmer's totient problem". Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 35: 43–49. ISSN 0138-9491. MR 2894552. Zbl 1240.11005.

External links

Weisstein, Eric W. "Lehmer's Totient Problem". MathWorld.

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