Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Berend 1998 arithmetical properties middle binomial coefficients
Berend, Daniel and Harmse, Jørgen E., On some arithmetical properties of middle binomial coefficients. Acta Arith. 84 (1998), no. 1, 31--41, doi:10.4064/aa-84-1-31-41 (Crossref record and the publisher's page). No copyright or license line appears in the file's text layer; the publisher's record (https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/84/1/110114/on-some-arithmetical-properties-of-middle-binomial-coefficients, read 2026-10-02) labels the download "Free download under CC-BY license", a Creative Commons Attribution license with no version or URL named; the site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the website, not the article.
The paper studies the distribution of the middle binomial coefficients C(2n,n) modulo prime powers. Theorem 1.1 shows that for every odd prime power p^e and every residue class s mod p^e there are infinitely many n with C(2n,n) congruent to s, equivalently that the sequence is dense in the ring of p-adic integers; Remark 1.1 notes this fails for powers of 2 since C(2n,n) is always even. Theorem 1.2 strengthens this modulo a prime: for every odd prime p the sequence C(2n,n) is weakly well-distributed modulo p, meaning that among the n in a window M <= n <= N with C(2n,n) prime to p, each non-zero residue class mod p takes a share tending to 1/(p - 1) as N - M grows (Definition 1.1, p. 32), while the authors explain in Remark 1.2 that their method does not reach prime powers because the Lucas-type congruence for prime moduli has no simple analog mod p^e. Theorem 1.2 is deduced from Theorem 1.3, a general well-distribution result for sequences a_n = alpha_{d_0} alpha_{d_1} ... alpha_{d_k} formed from the base-r digits of n in a compact group G, under the hypotheses that the closed subgroup generated by alpha_0, ..., alpha_{r-1} is G and that these elements do not all lie in one coset of a proper closed normal subgroup. The introduction records that Erdos's conjecture that C(2n,n) is never squarefree for n > 4 was settled by Granville-Ramare and by Velammal. For Erdos problem 376, whether infinitely many C(2n,n) are coprime to 105, the paper's theorems treat one prime at a time: Theorem 1.1 gives infinitely many n with C(2n,n) prime to 3, and likewise for 5 and for 7, but nothing for two of these primes at once. Section 4 (p. 40) records the theorem of Erdos, Graham, Ruzsa and Straus that C(2n,n) is infinitely often prime to pq for distinct odd primes p and q, judges the paper's methods unlikely to apply to such joint questions, and ends with Graham's question whether (C(2n,n), 105) = 1 for infinitely many n, which it says carries a prize (a personal communication of Graham).
Bears on. #376
Results to transcribe.
- Theorem 1.1: For every odd prime power p^e and every residue class s mod p^e there are infinitely many n with C(2n,n) in p^e Z + s; equivalently (C(2n,n)) is dense in Z_p.
- Theorem 1.2: For every odd prime p, the sequence C(2n,n) is weakly well-distributed modulo p.
- Theorem 1.3: In a compact group G, the digit-product sequence a_n = alpha_{d_0}...alpha_{d_k} built from the base-r digits of n is well-distributed provided the closed subgroup generated by alpha_0, ..., alpha_{r-1} is G and these elements do not all lie in one coset of a proper closed normal subgroup.