What does this research mean for the field?

The Erdős Square-Free Binomial Coefficient Conjecture, which asserts that the central binomial coefficient C(2n, n) is never square-free for n > 4, is fully validated through a 100% machine-checked formal verification in Lean 4. Novelty: ClaimNovelty.CONFIRMATORY. Consensus alignment: ConsensusAlignment.SUPPORTS_CONSENSUS.

What question did this study set out to answer?

Resolve and provide a formal, machine-checked verification of the Erdős Square-Free Binomial Coefficient Conjecture, demonstrating that central binomial coefficients are never square-free for integers n > 4.

August 29, 2026Open Access

On the Square Factors of Central Binomial Coefficients

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Resolve and provide a formal, machine-checked verification of the Erdős Square-Free Binomial Coefficient Conjecture, demonstrating that central binomial coefficients are never square-free for integers n > 4.
Synthesized Kummer's carry theorem on p-adic valuations with Granville and Ramaré's analytic prime sum estimates across medium-range primes.
Formally implemented and verified the square-free factor predicates, central binomial evaluations, and explicit prime-square factor certificates using the Lean 4 proof assistant and Mathlib with zero axioms or missing steps.
Confirmed the complete exception set where central binomial coefficients remain square-free strictly at n ∈ {1, 2, 4}, providing explicit square-factor certificates for all other small values (such as 3² dividing C(6,3)).
Demonstrated that medium-range prime densities in the interval √(2n) < p ≤ √(8n/3) guarantee a base-p carry, proving that square factors unconditionally exist for all n > 4.

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Implication

Theoretical analysis demonstrates non-trivial square factors in central binomial coefficients for integers exceeding four, highlighting the validity of the Erdős conjecture via formal proof.

Cite This Study

Charles EDOU NZE (2026) studied this question.

synapsesocial.com/papers/6a92996e8e5d7d1fc0c116d8 https://doi.org/10.5281/zenodo.22133078

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Implication

Theoretical analysis demonstrates non-trivial square factors in central binomial coefficients for integers exceeding four, highlighting the validity of the Erdős conjecture via formal proof.

On the Square Factors of Central Binomial Coefficients

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Discussion

Implication

Cite This Study

On the Square Factors of Central Binomial Coefficients

Key Points

Discussion

Implication

Cite This Study

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