We prove new estimates for the remainder terms in the known asymptotic formulas for three famous problems, by using the contemporary bounds for triple exponential sums.

Let the \emph{Fleck numbers}, $C_n(t,q)$, be defined such that \[ C_n(t,q)=\sum_{k\equiv q (mod n)}(-1)^k\binom{t}{k}. \] For prime $p$, Fleck obtained the result $C_p(t,q)\equiv 0 (mod p^{{\left \lfloor (t-1)/(p-1)\right \rfloor}}} )$, where $\lfloor.\rfloor$ denotes the usual floor function. This congruence was extended 64 years later by Weisman, in 1977, to include the case $n=p^\alpha$. In this paper we show that the Fleck numbers occur naturally when one considers a symmetric $n\times n$ matrix, $M$, and its inverse under matrix multiplication. More specifically, we take $M$ to be a symmetrically constructed $n\times n$ associated magic square of odd order, and then consider the reduced coefficients of the linear expansions of the entries of $M^t$ with $t\in \mathbb{Z}$. We also show that for any odd integer, $n=2m+1$, $n\geq 3$, there exist geometric polynomials in $m$ that are linked to the Fleck numbers via matrix algebra and $p$-adic interaction. These polynomials generate numbers that obey a reciprocal type of congruence to the one discovered by Fleck. As a by-product of our investigations we observe a new identity between values of the Zeta functions at even integers. Namely \[ \zeta{(2j)}=(-1)^{j+1}\left (\frac{j\pi^{2j}}{(2j+1)!}+\sum_{k=1}^{j-1}\frac{(-1)^k\pi^{2j-2k}}{(2j-2k+1)!}\zeta{(2k)}\right ). \] We conclude with examples of combinatorial congruences, Vandermonde type determinants and Number Walls that further highlight the symmetric relations that exist between the Fleck numbers and the geometric polynomials.

Let $(a,b,c)$ be a primitive Pythagorean triple such that $a^2+b^2=c^2$ with even $b$. In 1956, L. Jeśmanowicz conjectured that the equation $a^x+b^y=c^z$ has only the solution $(x,y,z)=(2,2,2)$ in positive integers. In this paper, we give various new results on this conjecture. In particular, we prove that if the equation has a solution $(x,y,z)$ with even $x,z$ then $x/2$ and $z/2$ are odd.

Given an integer $d\ge 2$, a $d$-{\it normal number}, or simply a {\it normal number}, is a real number whose $d$-ary expansion is such that any preassigned sequence, of length $k\ge 1$, of base $d$ digits from this expansion, occurs at the expected frequency, namely $1/d^k$. We construct large families of normal numbers using classified prime divisors of integers.

A well-known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is quite out of reach at present, even under the assumption of the Riemann Hypothesis. The aim of this paper is to provide the upper bounds for the exceptional set for this conjecture under the assumption of some heuristic hypotheses.

In this paper we prove the arbitrary potential modularity for an elliptic curve defined over a totally real number field.

We prove a conjecture posed in [11] and continue the process of determining Euler-Rabinowitsch polynomials that produce consecutive primes in a given range of inputs, and the relationship with class numbers of the underlying quadratic field.

We treat Hermite's formulas for $q$-analogues of the Hurwitz zeta function. As their application, we study the classical limit of modified $q$-analogues of the Hurwitz zeta function. We also treat $q$-analogues of the Milnor multiple gamma function.