Lander, Parkin, and Selfridge Conjecture

formal-conjectures · wikipedia · ams-11

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 $k \leq n + m$.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://en.wikipedia.org/wiki/Lander,_Parkin,_and_Selfridge_conjecture
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/LanderParkinAndSelfridgeConjecture.lean
Local target: Corpus.WikipediaLanderParkinAndSelfridgeConjecture.lander_parkin_selfridge
Source SHA-256: 1d5fc2be2670b2032b680f19afb432bfc2a0c56a3c08d3975eed8a78ee5b6c09
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record