Consecutive integer blocks each with product $\equiv 1 \pmod p$: possible for every $k$? (Erdős #1056)
Statement
Fix $k\geq 2$. Do there exist a prime $p$ and consecutive intervals $I_1,I_2,\ldots,I_k$ of positive integers — each a block of consecutive integers, with $I_{i+1}$ beginning immediately after $I_i$ ends — such that $$\prod_{n\in I_i} n \equiv 1 \pmod{p}\qquad\text{for every } 1\leq i\leq k?$$ The question is whether such a configuration exists for every $k$.
Acceptance. FULLY RESOLVES (OPEN, proof-shaped): a proof that for every $k\geq 2$ such a prime and intervals exist (or a proof that some $k$ admits none). ADVANCES: exhibit an explicit witness for a value of $k$ with no previously known example (currently every $k\geq 4$ is open) — give the prime $p$ and the interval endpoints, and verify each block product $\equiv 1\pmod p$; equivalently, a prime $p$ and $k+1$ integers with equal factorial residues modulo $p$. Each such witness is fully machine-checkable and pushes the frontier past the current best (state it in words: $k=3$, Makowski). Deliver the witness (prime, endpoints, per-block residue checks) and/or the search program together with the largest $k$ attained.
Background
This is problem A15 in Guy's Unsolved Problems in Number Theory [Gu04]; listed as open on erdosproblems.com/1056 (fetched 2026-07-21, status 'open'). Known cases: in a 1979 letter Erdős settled $k=2$ via $3\cdot 4\equiv 5\cdot 6\cdot 7\equiv 1\pmod{11}$, and Makowski [Ma83] settled $k=3$ via $2\cdot 3\cdot 4\cdot 5\equiv 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\equiv 12\cdot 13\cdot 14\cdot 15\equiv 1\pmod{17}$. No example is known for $k\geq 4$. Noll and Simmons posed the more general question of whether $q_1!\equiv\cdots\equiv q_k!\pmod p$ (with $q_1<\cdots<q_k$) is solvable for arbitrarily large $k$ — which is equivalent, since a block product $\prod_{n\in I_i}n$ equals a ratio of factorials modulo $p$, so the requirement is exactly that the running factorial hits one residue class at $k+1$ successive cut points. This connects to the distribution of factorial residues studied in Erdős #478 (erdosproblems.com/478; also on the SciNet venue), which asks whether the number of distinct values of $m!\bmod p$ approaches $(1-1/e)p$. OEIS A060427 records associated data. A formalised statement exists in the google-deepmind/formal-conjectures Lean repository. No cash prize. Attacker's tool: for each candidate $k$, scan primes $p$ and compute the partial products $1,\,1\cdot 2,\,1\cdot 2\cdot 3,\ldots \bmod p$, looking for a residue value returned to at least $k+1$ times — a direct finite search that could exhibit the first $k=4$ (and beyond) witness.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #1056 (T. F. Bloom) | website |
| REF-02 | OEIS A060427 — sequence associated with Erdős Problem #1056 | website |
| REF-03 | Lean formalisation (google-deepmind/formal-conjectures) of Erdős #1056 | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.