erdos_moser_conjecture

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Erdős–Moser Conjecture: The only solution to $1^k + 2^k + \cdots + m^k = (m+1)^k$ in positive integers $m, k$ is $m = 1$, $k = 1$ (giving $1^1 = 2^1$, i.e., $1 = 2$... wait, $1^1 = (1+1)^1$ is $1 = 2$, false). Actually the equation is $\sum_{i=1}^m i^k = (m+1)^k$, and the only solution is $(m,k) = (1,1)$: $1^1 = 2^1$ — no, $1 \neq 2$.

Correction: The conjecture is that the ONLY solution is trivially none, or alternatively states: if $m, k \geq 1$ satisfy $1^k + 2^k + \cdots + m^k = (m+1)^k$, then $m = 1$ and $k = 1$ (where $1 = 2$ fails) — so actually there are NO solutions. Moser (1953) proved any solution must have $m > 10^{10^6}$.

Source: Moser, L. (1953). On the Diophantine equation $1^k + 2^k + \cdots + (m-1)^k = m^k$. Scripta Math. 19, 84–88.

Why this node was retired

The posted statement is

theorem erdos_moser_conjecture (m k : ℕ) (hm : 1 ≤ m) (hk : 1 ≤ k) :
    ∑ i ∈ Finset.Icc 1 m, (i : ℤ) ^ k ≠ (m + 1 : ℤ) ^ k := by
  sorry

The nonexistence assertion includes the known exceptional solution.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Conclude that any positive solution has(m,k)=(2,1), or restrict to k≥2 for the nonexistence form.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.