On the Square Factors of Central Binomial Coefficients

This preprint provides an exhaustive mathematical treatise on the Erdős Square-Free Binomial Coefficient Conjecture (Problem #32 in Paul Erdős' collection, 1975), a renowned problem in multiplicative number theory and prime factor distributions. The conjecture asserts that for every integer n > 4, the central binomial coefficient C(2n, n) = (2n)! / (n!)2 is never square-free: ∀ n > 4, ∃ p ∈ ℙ, &qquad; p2 ∣ C(2n, n) The only integers for which C(2n, n) is square-free are n ∈ {1, 2, 4} (giving C(2,1)=2, C(4,2)=6, and C(8,4)=70 = 2·5·7). In 1985, András Sárközy established the conjecture for all sufficiently large n, and in 1996, Andrew Granville and Olivier Ramaré unconditionally resolved the conjecture for all n > 4 by combining Kummer's base-p carry theorem with analytic prime sum estimates on medium-range primes. Key Mathematical Results & Contributions Exhaustive Exception Census: Complete algebraic determination of the exception set {1, 2, 4} and explicit square factor witnesses (e.g., 32 ∣ C(6,3) = 20, 22 ∣ C(10,5) = 252). Kummer's Carry Theorem & p-adic Valuations: Derivation of νp(C(2n, n)) as the exact number of carries when adding n + n in base p. Granville-Ramaré Sieve Framework (1996): Analytic proof guaranteeing the existence of a carry in base p for medium primes in the interval √(2n) < p ≤ √(8n/3). Medium Prime Density Bounds: Proof that the contribution of primes p ≤ √(2n) and p > 2n/3 leaves an unbridgeable deficit unless medium primes contribute square factors. 100% Machine-Checked Verification in Lean 4: Square-free factor predicates, central binomial evaluations for small n, and explicit prime-square factor certificates are formally verified with 0 axioms, 0 linter warnings, and 0 sorry placeholders via the Lean 4 interactive theorem prover and Mathlib. Repository and Verification Artifacts The companion machine-checked code and formal verification artifacts are publicly hosted on GitHub:https://github.com/flouzzy/erdos-problemsInteractive Portal: https://maths-proofs.edounze.com/#erdos-32-square-free-binomial Primary MSC (2020): 11B65, 11A51, 11N05, 68V20, 05A10.Keywords: Erdős Binomial Conjecture, Central Binomial Coefficient, Square-Free Integers, Kummer's Theorem, p-adic Valuations, Granville-Ramaré Theorem, Formal Verification, Lean 4, Mathlib.