OP-00008 Open

Pillai's conjecture on axⁿ − byᵐ = c

Let $a$, $b$ and $c$ be positive integers. The equation $a x^n - b y^m = c$ has only finitely many solutions in integers $x, y, m, n \ge 2$ with $(m, n) \ne (2, 2)$.

For $a = b = 1$ this says that the gaps between consecutive perfect powers tend to infinity.

Source: S. S. Pillai, J. Indian Math. Soc. 2 (1936) and Bull. Calcutta Math. Soc. 37 (1945), for a = b = 1; the form with coefficients is a later generalisation, see Waldschmidt, arXiv:0908.4031.

Classification

Primary: NT — Number Theory
Keywords: exponential Diophantine equations, perfect powers

References

@article{pillai1936,
  author = {Pillai, S. S.},
  title = {On $a^x - b^y = c$},
  journal = {Journal of the Indian Mathematical Society (N.S.)},
  volume = {2},
  pages = {119--122},
  year = {1936}
}

@article{pillai1945,
  author = {Pillai, S. S.},
  title = {On the equation $2^x - 3^y = 2^X + 3^Y$},
  journal = {Bulletin of the Calcutta Mathematical Society},
  volume = {37},
  pages = {15--20},
  year = {1945}
}

@misc{waldschmidt2009,
  author = {Waldschmidt, Michel},
  title = {Perfect powers: {P}illai's works and their developments},
  eprint = {0908.4031},
  archivePrefix = {arXiv},
  year = {2009}
}

@misc{wikipedia,
  title = {Catalan's conjecture},
  howpublished = {Wikipedia},
  url = {https://en.wikipedia.org/wiki/Catalan%27s_conjecture}
}

@misc{formalconjectures,
  author = {{The Formal Conjectures Authors}},
  title = {Formal Conjectures},
  url = {https://github.com/google-deepmind/formal-conjectures/blob/0771383387505c96d1b2f6a3d35088ad00892c5c/FormalConjectures/Wikipedia/Catalan.lean#L42},
  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_00008.

/-- OP-00008: Pillai's conjecture on axⁿ − byᵐ = c -/
def Problem_OP_00008 : Prop :=
  ∀ (a b c : ℕ),
      0 < a →
        0 < b →
          0 < c →
            {(x, y, m, n) |
                1 < x ∧ 1 < y ∧ 1 < m ∧ 1 < n ∧ (m, n) ≠ (2, 2) ∧ a * x ^ n - b * y ^ m = c}.Finite

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_00008

theorem resolution : RegistryProblems.Problem_OP_00008 := by
  sorry

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

[[statements]]
label = "Theorem 1"
lean = "resolution"
latex = "..."
problem = "OP-00008"
claim = "full"

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

theorem disproof : ¬ RegistryProblems.Problem_OP_00008 := 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-00008, "Pillai's conjecture on axⁿ − byᵐ = c", clopen.solutions.