Hwang 2024 frobenius problem numerical semigroups generated binomial

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corollary_3_3: Hwang and Song's corollary that, for n with at least two distinct prime factors, gives the Apéry set of the binomial semigroup with respect to n, its Frobenius number and genus, and shows the Frobenius number is its only pseudo-Frobenius number, so the semigroup has type 1.

corollary_3_7: Hwang and Song's prime-power case: for n = p^m the semigroup generated by the C(n,k)/p is minimally generated by the C(p^m,p^i)/p for i below m, with embedding dimension m, and its Apéry set, Frobenius number, genus and single pseudo-Frobenius number are given in closed form.

lemma_2_2: Hwang and Song's congruence: if p is one of the primes of n and k is at most its exponent in n, then the binomial coefficient C(n, p^k) is congruent to n/p^k modulo n.

theorem_0_1: Hwang and Song's main theorem: for n with at least two distinct prime factors the semigroup generated by C(n,1), ..., C(n,n-1) has Frobenius number the sum of (p-1) C(n,p^j) over the prime powers p^j dividing n, minus n, with a prime-power analogue after division by p.

theorem_3_2: Hwang and Song's theorem that for n with at least two distinct prime factors the semigroup generated by C(n,1), ..., C(n,n-1) is generated by C(n,1) together with C(n,p_i^j) for the prime powers p_i^j dividing n, so that its embedding dimension is one plus the sum of the exponents.

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WonTae Hwang, Kyunghwan Song, The Frobenius problem for Numerical Semigroups generated by binomial coefficients. arXiv:2412.17882 (2024). The copy read for this card is arXiv v3 (3 October 2025), whose reference [1], the Erdős problem's web page, records its last access as 1 October 2025.

The paper determines the Apéry set and the Frobenius number of the numerical semigroup S(B_n) generated by the binomial coefficients C(n,1), ..., C(n,n-1), divided by p when n is a power of a prime p (Ram showed their gcd is p there and 1 otherwise). Theorem 0.1(1) (p. 2) covers n = p_1^{k_1} ... p_t^{k_t} with t >= 2 distinct primes and gives F(S(B_n)) as the sum over i and 1 <= j <= k_i of (p_i - 1) C(n, p_i^j), minus n; Theorem 0.1(2) (p. 3) covers n = p^m, m >= 1, giving F(S(B_n)) = ((p-1)/p) sum_{i=1}^{m-1} C(p^m, p^i) minus p^{m-1}. The print calls the k_i of part (1) nonnegative integers; the result pages read them as positive. The method computes the Apéry set from a minimal generating set of S(B_n) and then uses the Apéry-set formula for the Frobenius number. The introduction (p. 3) points for the proof to Lemma 2.2 and to Theorems 3.2 and 3.7; the print's Section 3 labels the last one Corollary 3.7, and the paper omits the proofs of the prime-power case (p. 8). Section 3 also gives the minimal generators, embedding dimension, genus and pseudo-Frobenius numbers, and shows S(B_n) is telescopic, a complete intersection and of type 1. Remark 0.2 (p. 3) states that Theorem 0.1(1)(b) solves a problem of Erdős, citing the erdosproblems.com entry for Problem 435. Section 4 gives applications: nonnegative integer identities among binomial coefficients with the same upper index, and admissible pairs for (s, s+1, s+p)-core partitions attached to S(B_n).

Source: https://arxiv.org/abs/2412.17882. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2412.17882), every other right reserved.

Bears on.

• Problem 435: the problem asks, for n not a prime power, for the largest integer that is not a nonnegative integer combination of C(n,1), ..., C(n,n-1). Theorem 0.1(1)(b) gives that integer, F(S(B_n)), in closed form for every n with at least two distinct prime factors, and Remark 0.2 states that it solves the problem.

Read status: claims checked for Theorem 0.1, Remark 0.2, Lemma 2.2, Theorems 3.1, 3.2 and 3.6 and Corollaries 3.3, 3.4, 3.5, 3.7 and 3.8, read clause by clause on the page images of the print; the proofs of Lemma 2.2, Theorem 3.2 and Corollary 3.3 followed for structure only and not checked. Nothing here is independently reviewed.

Results.

• Theorem 0.1 (pp. 2-3) and Remark 0.2 (p. 3): the Apéry set and Frobenius number of S(B_n), for n with at least two distinct prime factors and for n = p^m.
• Lemma 2.2 (p. 5): C(n, p^k) is congruent to n/p^k modulo n when p is prime and p^k divides n.
• Theorem 3.2 (pp. 6-7): for n not a prime power, C(n,1) and the C(n, p_i^j) minimally generate S(B_n), with embedding dimension 1 + sum k_i.
• Corollary 3.3 (p. 7), with Corollaries 3.4 and 3.5 (p. 8): Apéry set, Frobenius number, genus and pseudo-Frobenius numbers for n not a prime power.
• Theorem 3.6 and Corollary 3.7 (pp. 8-9), with Corollary 3.8 and Remark 3.9 (p. 9): the same for n = p^m.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.