Pomerance 2026 remarks middle binomial coefficient

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Carl Pomerance, Remarks on the middle binomial coefficient. Integers 26 (2026), #A47. doi:10.5281/zenodo.19403814. No license line is printed in the file (p. 1 carries "#A47 INTEGERS 26 (2026)" and "DOI: 10.5281/zenodo.19403814"); the journal's home page states "All works of this journal are licensed under a Creative Commons Attribution 4.0 International License" (https://math.colgate.edu/~integers/, read 2026-10-02), the Creative Commons Attribution 4.0 license; the Zenodo deposit was not consulted.

Theorem 1 proves that for any fixed eta < 1/log 4 = 0.721..., a density-one set of integers m has (m+k)!/m! = (m+1)(m+2)...(m+k) dividing C(2m,m) for every positive k <= eta log m, strengthening an exercise following Theorem 2 of the author's earlier paper (where only fixed k was handled). Theorem 2 shows that with the product replaced by the binomial coefficient C(m+k,k), divisibility by C(2m,m) holds for a density-one set of m and all k <= exp(0.8 sqrt(log m)). The method is elementary: prime-by-prime valuation comparison via Kummer/Legendre-type carry counting, combined with a density argument excluding the sparse m whose binary or p-adic digits misbehave. The note situates itself against recent AI-assisted work on an Erdos problem (pointing to Sothanaphan's write-up) and recalls the work of Sanna and of Ford-Konyagin on when m divides C(2m,m). The note does not mention problem 400; Theorem 1 bears on it through a substitution made here: (m+k)!/m! dividing C(2m,m) means (m+k)! m! divides (2m)!, so with n = 2m, a_1 = m+k and a_2 = m the excess a_1 + a_2 - n is k, and g_2(n) >= floor(eta log(n/2)) for all but o(x) even n <= x.

Source: https://math.colgate.edu/~integers/aa47/aa47.pdf.

Bears on. #400

Results to transcribe.

• Theorem 1: For any fixed eta < 1/log 4 = 0.721..., a density-one set of m has (m+k)!/m! dividing C(2m,m) for all positive k <= eta log m.
• Theorem 2: For a density-one set of m, C(m+k,k) divides C(2m,m) for all positive k <= exp(0.8 sqrt(log m)).
• Context (earlier Theorem 2): The author's earlier result: for each fixed positive k, m+k divides C(2m,m) for a set of m of density 1.
• Context (Ford-Konyagin): The set of m with m | C(2m,m) has a density, slightly larger than 1/11, after Sanna's upper-density bound below 1/4 and the earlier paper's upper-density bound 1 - log 2.