A Greatest Common Divisor Criterion of Certain Binomial Coefficients

2026-06-22

AI Summary

This study resolves a long-standing conjecture from OEIS A080170 concerning the greatest common divisor of binomial coefficients: for \(k \geq 2\), define \(D(k) = \gcd_{2 \leq q \leq k+1} \binom{qk}{k}\); we prove that \(D(k) = 1\) if and only if \(n = k+1\) satisfies \(n/P > P\), where \(P\) denotes the largest prime power dividing \(n\). By integrating rigorous mathematical reasoning with formal verification, we provide the first complete proof of this criterion. The entire proof development and validation were automatically generated and checked within the Lean theorem prover using the MechMath AI agent. This result has been accepted into the Formal Conjectures project, thereby establishing the correctness of the number-theoretic criterion in all cases.

Abstract

The binomial greatest common divisor (gcd) criterion recorded as OEIS A080170 is proven. The criterion also appears as conjecture (17) in Ralf Stephan's list of OEIS conjectures. For $k\geq 2$, put \[ D(k)=\gcd_{2\leq q\leq k+1}\binom{qk}{k}, \qquad n=k+1. \] If $P$ is the largest prime-power component $p^a$ exactly dividing $n$, then the criterion asserts \[ D(k)=1 \quad\Longleftrightarrow\quad \frac{n}{P}>P. \] The proof is formalized in Lean and the Lean artifact is accepted as part of the Formal Conjectures project. Both the natural-language proof and the Lean formalization are generated by the MechMath Agent Team, an AI agent developed by the authors.

Problem

• binomial coefficients
• greatest common divisor
• prime-power
• number theory
• OEIS conjecture

Innovation

• binomial gcd criterion
• formal proof
• Lean
• AI-assisted theorem proving
• OEIS conjecture