Let $f(x)=ax^{2^l\cdot3^m}+b\in \mathbb{Z}[x]$ be a polynomial with $l\geq 1, l+m\geq 2, ab\neq 0$ and such that $f(k)\neq 0$ for any $k\geq 1$. We prove, under $ABC$ conjecture, that the product $\prod_{k=1}^n f(k)$ is not a $2^l\cdot3^m$-th power for $n$ large enough.

The main result of [23] on the non-existence of low-degree irreducible factors of the Cauchy-Liouville-Mirimanoff polynomials $E_p(x)$ for primes $p \equiv 2 \pmod{3}$ is extended to a similar result for $p \equiv 1 \pmod{3}$. We also give a partial result on the existence of higher-degree irreducible factors.

In this paper, we present some new identities providing relations between Ramanujan's cubic continued fraction $V(q)$ and the other three continued fractions $V(q^9)$, $V(q^{17})$ and $V(q^{19})$. In the process, we establish some new modular equations for the ratios of Ramanujan's theta functions. We also establish some general formulas for the explicit evaluations of ratios of Ramanujan's theta functions.

In this paper we introduce special generalized Bernoulli polynomials which generalize poly-Bernoulli polynomials and numbers. We call them multi-poly-Bernoulli polynomials and numbers. We prove a collection of important and fundamental identities satisfied by our multi-poly-Bernoulli polynomials and numbers.

For positive integers $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r}$ with $\alpha_{r} \geq 2$, the multiple zeta value or $r$-fold Euler sum is defined by \[\zeta(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r}) = \sum_{1\leq n_{1}<n_{2}<\cdots<n_{r}}n_1^{-\alpha_1}n_2^{-\alpha_2}\cdots n_r^{-\alpha_r}. \] where $|\alpha|=\alpha_1+\alpha_2+\cdots+\alpha_r$ and $r$ are the weight and depth of $\zeta(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r})$ respectively. By the general theorem given in [11], the multiple zeta value $\zeta(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r})$ can be expressed as a rational linear combination of products of multiple zeta values of lower depth if its depth and weight are of different parity. In other words, when the sum of its depth and weight is odd. However, there are still some exceptions for quadruple Euler sums. As conjectured in [6], a quadruple Euler sum with even weight exceeding 14 can be expressed as a rational linear combination of products of multiple zeta values of depth 1, 2 and 3 if and only if it is one of the following forms: $\zeta(1,a,b,a)$, $\zeta(b,1,a,a)$, $\zeta(b,b,1,a)$ or $\zeta(a,b,b,a)$ with $a=b$ or $b=1$. In this paper, we shall evaluate these quadruple Euler sums of even weight by the identities among multiple zeta values with variables and relation obtained from the stuffle formula of two multiple zeta values.

In this paper, by means of a combinatorial-geometric method, we give a new proof of Knopp's formula for Dedekind sums and its generalizations to multiple Dedekind sums attached to Dirichlet characters. The combinatorial-geometric method for studying Dedekind sums were introduced by Beck, who proved the well-known reciprocity formula for Dedekind sums and some of its generalizations by the method. The motive of this paper is to find a similar approch to Knopp's formula .

We present a hybrid approach to bounding exponential sums over $k$th powers via Vinogradov's mean value theorem, and derive estimates of utility for exponents $k$ of intermediate size.

In this paper, we show that the only positive integer solutions of the equation $2^x=x^2+y^2-2$ are $(x,y)=(3,1),~(5,3),~(7,9)$. We propose also the following conjecture: the equation $2^x=y^2+z^2(x^2-2)$, where $y,z$ are odd positive integers and $x$ is a positive integer such that $x^2-2$ is a prime number, has the only solutions $(x,y,z)=(3,1,1),~(5,3,1),~(7,9,1),~(13,3,7)$. The conjecture implies a recent result of Lee [4] which states that if $x^2-2$ is an odd prime number such that the class number $h(x^2-2)$ of the quadratic field $\mathbb{Q}[{\sqrt{x^2-2}}]$ is $1$, then $x=3,5,7,13$.

In this paper we consider genus one equations of degree $n$, namely a (generalised) binary quartic when $n=2$, a ternary cubic when $n=3$, and a pair of quaternary quadrics when $n=4$. A new definition for the minimality of genus one equations of degree $n$ over local fields is introduced. The advantage of this definition is that it does not depend on invariant theory of genus one curves. We prove that this definition coincides with the classical definition of minimality for all $n\le4$. As an application, we give a new proof for the existence of global minimal genus one equations over number fields of class number 1.

On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. It is shown that there are finitely many perfect powers in such a sequence whose first term is greater than $1$. Moreover, if the first term is divisible by $6$ and the generating point is triple another rational point then there are no perfect powers in the sequence except possibly an $l$th power for some $l$ dividing the order of $2$ in the first term.