OP-00010 Open

The Lander–Parkin–Selfridge conjecture

If $\sum_{i=1}^{n} a_i^k = \sum_{j=1}^{m} b_j^k$, where the $a_i$ and $b_j$ are positive integers with $a_i \ne b_j$ for all $i$ and $j$, then $n + m \ge k$.

Euler's sum of powers conjecture is the case $m = 1$. It is false for $k = 4$ and $k = 5$, but the known counterexamples still satisfy $n + m \ge k$.

Source: L. J. Lander, T. R. Parkin and J. L. Selfridge, A survey of equal sums of like powers, Math. Comp. 21 (1967), 446–459.

Classification

Primary: NT — Number Theory
Keywords: Diophantine equations, equal sums of like powers

References

@article{lps1967,
  author = {Lander, L. J. and Parkin, T. R. and Selfridge, J. L.},
  title = {A survey of equal sums of like powers},
  journal = {Mathematics of Computation},
  volume = {21},
  pages = {446--459},
  year = {1967}
}

@misc{wikipedia,
  title = {Lander, Parkin, and Selfridge conjecture},
  howpublished = {Wikipedia},
  url = {https://en.wikipedia.org/wiki/Lander,_Parkin,_and_Selfridge_conjecture}
}

@misc{formalconjectures,
  author = {{The Formal Conjectures Authors}},
  title = {Formal Conjectures},
  url = {https://github.com/google-deepmind/formal-conjectures/blob/0771383387505c96d1b2f6a3d35088ad00892c5c/FormalConjectures/Wikipedia/LanderParkinAndSelfridgeConjecture.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_00010.

/-- OP-00010: The Lander–Parkin–Selfridge conjecture -/
def Problem_OP_00010 : Prop :=
  ∀ (k n m : ℕ) (x : Fin n → ℕ) (y : Fin m → ℕ),
      0 < n →
        0 < m →
          (∀ (i : Fin n), 0 < x i) →
            (∀ (j : Fin m), 0 < y j) →
              (∀ (i : Fin n) (j : Fin m), x i ≠ y j) → ∑ i, x i ^ k = ∑ j, y j ^ k → k ≤ n + m

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_00010

theorem resolution : RegistryProblems.Problem_OP_00010 := by
  sorry

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

[[statements]]
label = "Theorem 1"
lean = "resolution"
latex = "..."
problem = "OP-00010"
claim = "full"

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

theorem disproof : ¬ RegistryProblems.Problem_OP_00010 := 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-00010, "The Lander–Parkin–Selfridge conjecture", clopen.solutions.