arXiv subjects
Publications and source records attributed to Bishnu Paudel.
We explore the dependence of the minimal integral Mahler measure of Galois quartic fields on the discriminant of the field. We obtain density results which are conditional on the ABC conjecture as well as several unconditional results.
We define $\overline{R_l^*}(n)$ as the number of overpartitions of $n$ in which non-overlined parts are not divisible by $l$. In a recent work, Nath, Saikia, and the second author established several families of congruences for $\overline{R_l^*}(n)$, with particular focus on the cases $l=6$ and $l=8$. In the concluding remarks of their paper, they conjectured that $\overline{R_6^*}(n)$ satisfies an infinite family of congruences modulo $128$. In this paper, we confirm their conjectures using elementary methods. Additionally, we provide elementary proofs of two congruences for $\overline{R_6^*}(n)$ previously proven via the machinery of modular forms by Alanazi, Munagi, and Saikia.
Recently, Alanazi, Munagi, and Saikia employed the theory of modular forms to investigate the arithmetic properties of the function $\overline{R_{\ell,\mu}}(n)$, which enumerates the overpartitions of $n$ where no part is divisible by either $\ell$ or $\mu$, for various integer pairs $(\ell, \mu)$. In this paper, we substantially extend several of their results and establish infinitely many families of new congruences. Our proofs are entirely elementary, relying solely on classical $q$-series manipulations and dissection formulas.
Let $T_{\ell,k}(n)$ denote the number of $\ell$-regular $k$-tuple partitions of $n$. In a recent work, Nath, Saikia, and Sarma derived several families of congruences for $T_{\ell,k}(n)$, with particular emphasis on the cases $T_{2,3}(n)$ and $T_{4,3}(n)$. In the concluding remarks of their paper, they conjectured that $T_{2,3}(n)$ satisfies an infinite set of congruences modulo 6. In this paper, we confirm their conjecture by proving a much more general result using elementary $q$-series techniques. We also present new families of congruences satisfied by $T_{\ell,k}(n)$.
We obtain a complete description of the integer group determinants for $\mathbb Z_{18}$ (these are the $18\times18$ circulant determinants with integer entries) and $\mathbb Z_3 \times \mathbb Z_6$, the two abelian groups of order 18. This completes the groups of order less than 20.
We consider the integer group determinants for groups that are semidirect products of $\mathbb Z_p$ and $\mathbb Z_n$ with $p$ prime and $n\mid p-1$. We give a complete description of the integer group determinants for the general affine groups of degree one GA(1,$p$) when $p=5,7,11$ and $23$, and for $\mathbb Z_7\rtimes \mathbb Z_3,$ $\mathbb Z_{11}\rtimes \mathbb Z_5$ and $\mathbb Z_{13}\rtimes \mathbb Z_6,$ showing that the obvious divisibility and congruence conditions arising from the form of the group determinant when $n=p-1$ or $\frac{1}{2}(p-1)$, can be sufficient as well as necessary for these types of groups (although in the latter case we must work with norms of integers in a quadratic field). For $p=13$ this also happens for the remaining groups of this type, $\mathbb Z_{13}\rtimes_5 \mathbb Z_4$ and $\mathbb Z_{13}\rtimes \mathbb Z_3$, (working in an appropriate cubic and quartic field).
We obtain a complete description of the integer group determinants for the non-abelian groups of order 18.
We obtain a complete description of the integer group determinants for SmallGroup(16,8), the semidihedral group of order 16. While this paper was in preparation, a complete descriptions for this group was independently obtained by Yuka Yamaguchi and Naoya Yamaguchi in [21] (and the other remaining group of order 16 in [20]). We offer our version here for anyone interested in a slightly different approach.
We obtain a complete description of the integer group determinants for SmallGroup(16,13), the central product of the dihedral group of order eight and cyclic group of order four. These values are the same as the integer group determinants for SmallGroup(16,11), the direct product of the dihedral group of order eight and cyclic group of order two. It was not previously known that the integer group determinants do not determine the group.
Let $\mathbb Z_n$ denote the cyclic group of order $n$. We show how the group determinant for $G= \mathbb Z_n \times H$ can be simply written in terms of the group determinant for $H$. We use this to get a complete description of the integer group determinants for $\mathbb Z_2 \times D_8$ where $D_8$ is the dihedral group of order 8, and $\mathbb Z_2 \times Q_8$ where $Q_8$ is the quaternion group of order 8.
We obtain a complete description of the integer group determinants for $Q_{16},$ the dicyclic or generalized quaternion group of order 16.
For an irrational real $α$ and $γ\not \in \mathbb Z + \mathbb Zα$ it is well known that $$ \liminf_{|n|\rightarrow \infty} |n| ||nα-γ|| \leq \frac{1}{4}. $$ If the partial quotients, $a_i,$ in the negative `round-up' continued fraction expansion of $α$ have $R:=\liminf_{i\rightarrow \infty}a_i$ odd, then the 1/4 can be replaced by $$ \frac{1}{4}\left(1-\frac{1}{R}\right)\left(1-\frac{1}{R^2}\right), $$ which is optimal. The optimal bound for even $R\geq 4$ was already known.
For a given irrational number $α$ and a real number $γ$ in $(0,1)$ one defines the two-sided inhomogeneous approximation constant \begin{equation*} M(α,γ):=\liminf_{|n|\rightarrow\infty}|n| ||nα-γ||, \end{equation*} and the case of worst inhomogeneous approximation for $α$ \begin{equation*} ρ(α):=\sup_{γ\notin\mathbb{Z}+α\mathbb{Z}}M(α,γ). \end{equation*} We are interested in lower bounds on $ρ(α)$ in terms of $R:=\liminf_{i\rightarrow\infty}a_i,$ where the $a_i$ are the partial quotients in the negative (i.e.\ the `round-up') continued fraction expansion of $α$. We obtain bounds for any $R\geq 3$ which are best possible when $R$ is even (and asymptotically precise when $R$ is odd). In particular when $R\geq 3$ $$ ρ(α)\geq \cfrac{1}{6\sqrt{3}+8}=\cfrac{1}{18.3923\dots}, $$ and when $R\geq 4$, optimally, $$ ρ(α) \geq \cfrac{1}{4\sqrt{3}+2}=\cfrac{1}{8.9282\ldots}. $$
We consider the values taken by $n\times n$ circulant determinants with integer entries when $n$ is the product of two distinct odd primes $p,q$. These correspond to the integer group determinants for $\mathbb Z_{pq}$, the cyclic group of order $pq$. We show that $p^2$ and $q^2$ are not determinants (more generally we show that the classic necessary divisibility conditions are never sufficient when $n$ contains at least two odd primes). We obtain a complete description of the integer group determinants for $\mathbb Z_{15}$ (the smallest unresolved group) and partial results for general $n=3p.$
We determine the minimal non-trivial integer group determinant for the dicyclic group of order $4n$ when $n$ is odd. We also discuss the set of all integer group determinants for the dicyclic groups of order $4p$.