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OP-10024Status unassessedNumber theory

Lander, Parkin, and Selfridge Conjecture

Lander, Parkin, and Selfridge Conjecture

MSC 11 · Number theory

Inferred from the community field; finer classification has not been reviewed.

Specialist source · status unassessedAs of 2026-09-10Source & evidence ↓
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Problem statement

lander_parkin_selfridge

The Lander–Parkin–Selfridge conjecture: if the sum of nn positive integer kk-th powers equals the sum of mm positive integer kk-th powers, with all values on the left distinct from all values on the right, then n+m≥kn + m \geq k.

Formally, for positive integers k,n,m∈Nk, n, m \in \mathbb{N} and sequences x:{0,…,n−1}→Nx : \{0, \ldots, n-1\} \to \mathbb{N} and y:{0,…,m−1}→Ny : \{0, \ldots, m-1\} \to \mathbb{N} with xi>0x_i > 0, yj>0y_j > 0, and xi≠yjx_i \neq y_j for all i,ji, j, if

∑i=0n−1xik=∑j=0m−1yjk,\sum_{i=0}^{n-1} x_i^k = \sum_{j=0}^{m-1} y_j^k,

then k≤n+mk \leq n + m.

lander_parkin_selfridge.variants.five_three

Special case of the Lander–Parkin–Selfridge conjecture: there is no solution in positive integers to

x15+x25+x35=y5.x_1^5 + x_2^5 + x_3^5 = y^5.

That is, for all x1,x2,x3,y∈Nx_1, x_2, x_3, y \in \mathbb{N} with x1,x2,x3,y>0x_1, x_2, x_3, y > 0,

x15+x25+x35≠y5.x_1^5 + x_2^5 + x_3^5 \neq y^5.

This corresponds to the case k=5k = 5, n=3n = 3, m=1m = 1 of the general conjecture, where n+m=4<5=kn + m = 4 < 5 = k would be required to yield a counterexample.

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Formal statement

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/-
Copyright 2025 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

import FormalConjecturesUtil

/-!
# Lander, Parkin, and Selfridge Conjecture

**Reference:** https://en.wikipedia.org/wiki/Lander,_Parkin,_and_Selfridge_conjecture
-/

namespace LanderParkinSelfridge

/-- The Lander–Parkin–Selfridge conjecture: if the sum of $n$ positive integer $k$-th powers
equals the sum of $m$ positive integer $k$-th powers, with all values on the left distinct from
all values on the right, then $n + m \geq k$.

Formally, for positive integers $k, n, m \in \mathbb{N}$ and sequences
$x : \{0, \ldots, n-1\} \to \mathbb{N}$ and $y : \{0, \ldots, m-1\} \to \mathbb{N}$
with $x_i > 0$, $y_j > 0$, and $x_i \neq y_j$ for all $i, j$, if
$$\sum_{i=0}^{n-1} x_i^k = \sum_{j=0}^{m-1} y_j^k,$$
then $k \leq n + m$. -/
@[category research open, AMS 11]
theorem lander_parkin_selfridge :
    ∀ (k n m : ℕ) (x : Fin n → ℕ) (y : Fin m → ℕ),
      0 < n → 0 < m →
      (∀ i, 0 < x i) → (∀ j, 0 < y j) →
      (∀ i j, x i ≠ y j) →
      ∑ i, x i ^ k = ∑ j, y j ^ k →
      k ≤ n + m := by
  sorry

/-- Special case of the Lander–Parkin–Selfridge conjecture: there is no solution in positive
integers to
$$x_1^5 + x_2^5 + x_3^5 = y^5.$$
That is, for all $x_1, x_2, x_3, y \in \mathbb{N}$ with $x_1, x_2, x_3, y > 0$,
$$x_1^5 + x_2^5 + x_3^5 \neq y^5.$$
This corresponds to the case $k = 5$, $n = 3$, $m = 1$ of the general conjecture,
where $n + m = 4 < 5 = k$ would be required to yield a counterexample. -/
@[category research open, AMS 11]
theorem lander_parkin_selfridge.variants.five_three :
    ∀ x₁ x₂ x₃ y : ℕ,
      0 < x₁ → 0 < x₂ → 0 < x₃ → 0 < y →
      x₁ ^ 5 + x₂ ^ 5 + x₃ ^ 5 ≠ y ^ 5 := by
  sorry

end LanderParkinSelfridge

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Background

Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.

Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 2 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.

Copyright The Formal Conjectures Authors. Code: Apache-2.0; other original material: CC BY 4.0. Third-party sources retain their own licenses. Original source files and notices preserved.

References
  1. [1]Formal Conjectures — Lander, Parkin, and Selfridge Conjecture link
  2. [2]Original reference cited in the formalization link

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