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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. A preprint from February 2023 claims to resolve the conjecture in the positive.

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 > 10 and ω(n) ≥ 14.
• In 1988, Hagis showed that if 3 divides any solution n, then n > 10 and ω(n) ≥ 298848. This was subsequently improved by Burcsi, Czirbusz, and Farkas, who showed that if 3 divides any solution n, then n > 10 and ω(n) ≥ 40000000.
• A result from 2011 states that the number of solutions to the problem less than ${\displaystyle X}$ is at most ${\displaystyle {X^{1/2}/(\log X)^{1/2+o(1)}}}$.
• In an essentially two page preprint from 2023 Said Zriaa uses elementary properties about primes and Euler's totient function from number theory and elementary symmetric polynomials from algebra to give a short proof of Lehmer's totient conjecture.

References

1. Lehmer (1932)
2. ^ Zriaa, Said (2023), A new characterization of prime numbers and solution to Lehmer's conjecture on Euler's totient function, arXiv:2302.07368
3. Sándor et al (2006) p.23
4. Guy (2004) p.142
5. 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)
6. Luca and Pomerance (2011)

• 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. ISSN 0028-9825. Zbl 0436.10002.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• 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 Series. 6 (3): 255–261. ISSN 0028-9825. Zbl 0668.10006.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• 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.
• Luca, Florian; Pomerance, Carl (2011). "On composite integers n for which ${\displaystyle \varphi (n)\mid n-1}$". Bol. Soc. Mat. Mexicana. 17 (3): 13–21. ISSN 1405-213X. MR 2978700.
• 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" (PDF). Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 35: 43–49. ISSN 0138-9491. MR 2894552. Zbl 1240.11005.

• Conjectures
• Unsolved problems in number theory
• Multiplicative functions