Congruences modulo powers of 2 for generalized Frobenius partitions with six colors

Cui, Su-Ping; Gu, Nancy S. S.

doi:10.1142/s1793042119500647

Cited by 4 publications

(3 citation statements)

References 22 publications

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“…While there exists an extensive literature on the subject of congruences satisfied by generalised Frobenius partition functions, our focus in this note will be on parity results. We highlight here that a number of authors have proven congruence results with even moduli for these functions (see, for example, Andrews [1, Theorem 10.2], Baruah and Sarmah [2, 3], Chan et al [4], Cui and Gu [5], Cui et al [6] and Jameson and Wieczorek [9]). Several additional papers involving congruence results for generalised Frobenius partitions, but with odd moduli, also appear in the literature.…”

Section: Introductionmentioning

confidence: 93%

“…where the last statement follows from Theorem 2.1. Since (q p ; q p ) ∞ is a function of q p , and since we are interested in the parity of the values φ p −1 (pn + r) where 0 < r < p, we simply need to determine when pn + r = 3k 2 − k 2 for some integer k. After completing the square, this is equivalent to asking whether 24r + 1 ≡ 36k 2 − 12k + 1 (mod p) ≡ (6k − 1) 2 (mod p) [5] However, we assumed that 24r + 1 is a quadratic nonresidue modulo p in the statement of this theorem. Therefore, pn + r can never be represented as (3k 2 − k)/2 for any integer k. This implies that φ p −1 (pn + r) ≡ 0 (mod p).…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

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On the Parity of the Generalised Frobenius Partition Functions

Andrews¹,

Sellers

SOUFAN³

2022

Bull. Aust. Math. Soc.

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Andrews [Generalized Frobenius Partitions, Memoirs of the American Mathematical Society, 301 (American Mathematical Society, Providence, RI, 1984)] defined two families of functions, $\phi _k(n)$ and $c\phi _k(n),$ enumerating two types of combinatorial objects which he called generalised Frobenius partitions. Andrews proved a number of Ramanujan-like congruences satisfied by specific functions within these two families. Numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter $k.$ Our goal is to identify an infinite family of values of k such that $\phi _k(n)$ is even for all n in a specific arithmetic progression; in particular, we prove that, for all positive integers $\ell ,$ all primes $p\geq 5$ and all values $r, 0 < r < p,$ such that $24r+1$ is a quadratic nonresidue modulo $p,$ $$ \begin{align*} \phi_{p\ell-1}(pn+r) \equiv 0 \pmod{2} \end{align*} $$ for all $n\geq 0.$ Our proof of this result is truly elementary, relying on a lemma from Andrews’ memoir, classical q-series results and elementary generating function manipulations. Such a result, which holds for infinitely many values of $k,$ is rare in the study of arithmetic properties satisfied by generalised Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question.

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Section: Introductionmentioning

confidence: 93%

Section: Proof Of Theorem 13mentioning

confidence: 99%

On the Parity of the Generalised Frobenius Partition Functions

Andrews¹,

Sellers

SOUFAN³

2022

Bull. Aust. Math. Soc.

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“…While there exists an extensive literature on the subject of congruences satisfied by generalized Frobenius partition functions, our focus in this note will be on parity results. We highlight here that a number of authors have proven congruence results with even moduli for these functions; see, for example, the work of Andrews [1, Theorem 10.2], Baruah and Sarmah [2,3], Chan, Wang, and Yang [4], Cui and Gu, [5], Cui, Gu, and Huang [6], and Jameson and Wieczorek [9] where specific congruence results with even moduli are proved. Several additional papers involving congruence results for generalized Frobenius partitions, but with odd moduli, also appear in the literature.…”

Section: Introductionmentioning

confidence: 93%

On the Parity of the Generalized Frobenius Partition Functions $ϕ_k(n)$

Andrews¹,

Sellers²,

Soufan³

2022

Preprint

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In his 1984 Memoir of the American Mathematical Society, George Andrews defined two families of functions, φ k (n) and cφ k (n), which enumerate two types of combinatorial objects which Andrews called generalized Frobenius partitions. As part of that Memoir, Andrews proved a number of Ramanujan-like congruences satisfied by specific functions within these two families. In the years that followed, numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter k. In this brief note, our goal is to identify an infinite family of values of k such that φ k (n) is even for all n in a specific arithmetic progression; in particular, our primary goal in this work is to prove that, for all positive integers ℓ, all primes p ≥ 5, and all values r, 0 < r < p, such that 24r + 1 is a quadratic nonresidue modulo p, φ pℓ−1 (pn + r) ≡ 0 (mod 2) for all n ≥ 0. Our proof of this result is truly elementary, relying on a lemma from Andrews' Memoir, classical q-series results, and elementary generating function manipulations. Such a result, which holds for infinitely many values of k, is rare in the study of arithmetic properties satisfied by generalized Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question.

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