Let ${\overline{p}_{3}}(n)$ be the number of overpartition triples of $n$. By elementary series manipulations, we establish some congruences for ${\overline{p}}_{3}(n)$ modulo small powers of 2, such as
\[{\overline{p}_{3}}(16n+14)\equiv 0 \pmod{32}, \quad {\overline{p}_{3}}(8n+7)\equiv 0 \pmod{64}.\] We also find many arithmetic properties for ${\overline{p}_{3}}(n)$ modulo 7, 9 and 11, involving the following infinite families of Ramanujan-type congruences: for any integers $α\ge 1$ and $n \ge 0$, we have ${\overline{p}_{3}}\big({{3}^{2α+1}}(3n+2)\big)\equiv 0$ (mod $9\cdot 2^4$), $\overline{p}_{3}(4^{α-1}(56n+49)) \equiv 0$ (mod 7) and \[{\overline{p}_{3}}\big({{7}^{2α+1}}(7n+3)\big)\equiv {\overline{p}_{3}}\big({{7}^{2α+1}}(7n+5)\big)\equiv {\overline{p}_{3}}\big({{7}^{2α+1}}(7n+6)\big)\equiv 0 \pmod{7}.\]
Recently, Shen (2016) and Alanazi et al. (2016) studied the arithmetic properties of the $\ell$-regular overpartition function $\overline{A}_\ell (n)$, which counts the number of overpartitions of $n$ into parts not divisible by $\ell$. In this note, we will present some new congruences modulo $5$ when $\ell$ is a power of $5$.
Arithmetic Properties modulo powers of $2$ and $3$ for Overpartition $k$-Tuples with Odd Parts
Department of Mathematics, B. Borooah College（博罗阿学院数学系）；Mathematical and Physical Sciences division, School of Arts and Sciences, Ahmedabad University（艾哈迈达巴德大学文理学院数理科学部）；Department of Basic Sciences and Humanities, Assam Skill University（阿萨姆技能大学基础科学与人文学院）
In 1939, H. S. Zuckerman provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the overpartition function $\overline{p}(n)$. Computing $\overline{p}(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we provide a formula to compute the values of $\overline{p}(n)$ that requires only the values of $\overline{p}(k)$ with $k\leqslant n/2$. This formula is combined with a known linear homogeneous recurrence relation for the overpartition function $\overline{p}(n)$ to o…
Let $\mathrm{pod}_{-3}(n)$ denote the number of partition triples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\mathrm{pod}_{-3}(n)$ involving the following infinite family of congruences: for any integers $α\ge 1$ and $n\ge 0$, \[\mathrm{pod}_{-3}\Big({{3}^{2α+2}}n+\frac{23\times {{3}^{2α+1}}+3}{8}\Big)\equiv 0 \pmod{9}.\] We also establish some arithmetic relations between $\mathrm{pod}(n)$ and $\mathrm{pod}_{-3}(n)$, as well as some congruences for $\mathrm{pod}_{-3}(n)$ modulo 7 and 11.
Let ${{B}_{3}}(n)$ denote the number of partition triples of $n$ where each partition is 3-core. With the help of generating function manipulations, we find several infinite families of arithmetic identities and congruences for ${{B}_{3}}(n)$. Moreover, let $ω(n)$ denote the number of representations of a nonnegative integer $n$ in the form $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+3y_{1}^{2}+3y_{2}^{2}+3y_{3}^{2}$ with ${{x}_{1}},{{x}_{2}},{{x}_{3}},{{y}_{1}},{{y}_{2}},{{y}_{3}}\in \mathbb{Z}.$ We find three arithmetic relations between ${{B}_{3}}(n)$ and $ω(n)$, such as $ω(6n+5)=4{{B}_{3}}(6n+4).$
We introduce mod $l$ Milnor invariants of a Galois element associated to Ihara's Galois representation on the pro-$l$ fundamental group of a punctured projective line ($l$ being a prime number), as arithmetic analogues of Milnor invariants of a pure braid. We then show that triple quadratic (resp. cubic) residue symbols of primes in the rational (resp. Eisenstein) number field are expressed by mod $2$ (resp. mod $3$) triple Milnor invariants of Frobenius elements. For this, we introduce dilogarithmic mod $l$ Heisenberg ramified covering ${\cal D}^{(l)}$ of $\mathbb{P}^1$, which may be regarded…
In this article we study the Gross-Schoen diagonal cycle on a triple product of Shimura curves at a place of bad reduction. We relate the image of the diagonal cycle under the Abel-Jacobi map to certain period integral that governs the central critical value of the Garrett-Rankin type triple product $L$-function via level raising congruences. As an application we prove certain rank $0$ case of the Bloch-Kato conjecture for the symmetric cube motive of a weight $2$ modular form.
Let $A_{3}(n)$ (resp. ${{B}_{3}}(n)$) denote the number of partition pairs (resp. triples) of $n$ where each partition is 3-core. By applying Ramanujan's ${}_{1}ψ_{1}$ formula and Bailey's ${}_{6}ψ_{6}$ formula, we find the explicit formulas for $A_{3}(n)$ and $B_{3}(n)$. Using these formulas, we confirm a conjecture of Xia and establish many arithmetic identities satisfied by $A_{3}(n)$ and $B_{3}(n)$.
Let $Δ=Δ(a,b,c)$ be a hyperbolic triangle group, a Fuchsian group obtained from reflections in the sides of a triangle with angles $π/a,π/b,π/c$ drawn on the hyperbolic plane. We define the arithmetic dimension of $Δ$ to be the number of split real places of the quaternion algebra generated by $Δ$ over its (totally real) invariant trace field. Takeuchi has determined explicitly all triples $(a,b,c)$ with arithmetic dimension $1$, corresponding to the arithmetic triangle groups. We show more generally that the number of triples with fixed arithmetic dimension is finite, and we present an effici…
Problem 2 at the 56th International Mathematical Olympiad (2015) asks for all triples (a,b,c) of positive integers for which ab-c, bc-a, and ca-b are all powers of 2. We show that this problem requires only a primitive form of arithmetic, going back to the Pythagoreans, which is the arithmetic of the even and the odd.
