Erdős Problems · Nov 2026 · Finite computations

Erdős problems: finite checks, not new theorems

The Erdős Problems website (erdosproblems.com) lists open questions in number theory and combinatorics. We ran free, local computations on a few of them. Each computation covers a finite range. None of them proves or disproves a problem, and none is a new theorem.

Conceptual illustration generated with AI. Not experimental data or a depiction of the actual equipment.

Problem 699: central binomial coefficients

The statement we tested: for all 1 ≤ i < j ≤ n/2, there is a prime p ≥ i that divides both C(n,i) and C(n,j).

We scanned every n up to 255,667 and found no counterexample. The scan skips most pairs using a lemma we derived: for the largest prime q ≤ n, every pair with i > n - q is already covered by q, via Kummer's theorem. The proof is written out in our notes. We cross-checked the scanner against exact computation for small n and reproduced the known cases where the only valid prime equals i. As an independent check for this page, we recomputed all n ≤ 500 by brute force: no violations, and the equality cases at n = 10, 16, 28 and 244 match.

Problem 942: powerful numbers between consecutive squares

Let h(n) be the number of powerful numbers (if a prime p divides m, then p² divides m) in [n², (n+1)²). Erdős asked how fast h(n) can grow. We computed h(n) exactly for every n below 3.1 × 10^9. The largest value we found is h = 12, at n = 472,532,614 and three other values of n. For those we listed the 12 numbers in each interval and factored each one with a separate program: all are powerful and all lie in the interval.

Largest h(n) found so far, at the n where each new record first appears. Values for n up to 10^7 come from a full scan; the records 11 and 12 come from the extended scan below 3.1 × 10^9. Source: our computation logs, 9 November 2026. The data are consistent with slow growth, and say nothing about the constant in Erdős's question.

Smaller checks

Problem 288: an exact search over pairs of disjoint intervals with endpoints up to 1500 found only the three known-type solutions, and none new above 250. The search assumed disjoint intervals and capped sums below 3.
Problem 273: we read the state of the art on the site's forum and did not pursue it. Other people's searches are far ahead of our free two-core computer.

Limits

A finite check cannot settle a statement about all n.
These are our own runs. They are not peer-reviewed and nobody else has independently reproduced them.
We have not checked whether anyone has already covered these ranges.
Other computations from this line (for example #993 and #939) are partial and have no results to report yet.

Code and raw outputs are not publicly available.