OP-00006 Open

Lehmer's totient problem

Does there exist a composite number $n > 1$ such that Euler’s totient function $\varphi(n)$ divides $n - 1$?

Source: D. H. Lehmer, On Euler's totient function, Bull. Amer. Math. Soc. 38 (1932), 745–751. See Guy, Unsolved Problems in Number Theory, B37.

Classification

Primary: NT — Number Theory
Keywords: Euler's totient function

References

@article{lehmer1932,
  author = {Lehmer, D. H.},
  title = {On {E}uler's totient function},
  journal = {Bulletin of the American Mathematical Society},
  volume = {38},
  pages = {745--751},
  year = {1932}
}

@book{guy2004,
  author = {Guy, Richard K.},
  title = {Unsolved Problems in Number Theory},
  edition = {3},
  publisher = {Springer},
  year = {2004}
}

@misc{wikipedia,
  title = {Lehmer's totient problem},
  howpublished = {Wikipedia},
  url = {https://en.wikipedia.org/wiki/Lehmer%27s_totient_problem}
}

@misc{formalconjectures,
  author = {{The Formal Conjectures Authors}},
  title = {Formal Conjectures},
  url = {https://github.com/google-deepmind/formal-conjectures/blob/0771383387505c96d1b2f6a3d35088ad00892c5c/FormalConjectures/Wikipedia/LehmerTotient.lean#L30},
  note = {Lean statement under Apache-2.0; problem text under CC BY 4.0, or CC BY-SA 4.0 where it is based on Wikipedia}
}

Canonical Lean statement

A submission resolves this problem by proving the constant Problem_OP_00006.

/-- OP-00006: Lehmer's totient problem -/
def Problem_OP_00006 : Prop :=
  ∃ n > 1, ¬Prime n ∧ n.totient ∣ n - 1

Formalised by The Formal Conjectures Authors (Apache-2.0); reviewed by clemens-admin.

Available to match against in: Lean v4.33.1 with Mathlib v4.33.1.

Resolving this problem in Lean

Add the problems package as a dependency, import this problem's module, and prove the constant. Any published revision works.

In lakefile.toml:

[[require]]
name = "RegistryProblems"
git = "https://github.com/cthalhammer/registry-problems"
rev = "<published revision>"

In your proof:

import RegistryProblems.OP_00006

theorem resolution : RegistryProblems.Problem_OP_00006 := by
  sorry

In preprint.toml, mapping the claim to the declaration that establishes it:

[[statements]]
label = "Theorem 1"
lean = "resolution"
latex = "..."
problem = "OP-00006"
claim = "full"

To disprove it instead, prove the negation and claim "disproof", or "counterexample" if the proof exhibits one:

theorem disproof : ¬ RegistryProblems.Problem_OP_00006 := by
  sorry

You can also write the statement out in full; the verifier checks it matches up to definitional equality. Partial results and reductions are not matched against this statement.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00006, "Lehmer's totient problem", clopen.solutions.