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We study arithmetic progressions of squares over quadratic extensions of number fields. Using a method inspired by an approach of Mordell, we characterize such progressions as quadratic points on a genus $5$ curve. Specifically, we determine the set of $K$-quadratic points on this curve under certain conditions on the base field $K$. Our main results rely on the algebraic properties of specific elliptic curves after performing a base change to suitable number fields. As a consequence, we establish that, under appropriate assumptions, any non-elementary arithmetic progression of five or six squares properly defined over a quadratic extension of $K$ must be of a specific form. Moreover, we prove the non-existence of such progressions of length greater than six under these assumptions.

Let $K$ be a number field of degree $d$ so that $K/\mathbb Q$ is a Galois extension. The {\it normal basis theorem} states that $K$ has a $\mathbb Q$-basis consisting of algebraic conjugates, in fact $K$ contains infinitely many such bases. We prove an effective version of this theorem, obtaining a normal basis for $K/\mathbb Q$ of bounded Weil height with an explicit bound in terms of the degree and discriminant of $K$. In the case when $d$ is prime, we obtain a particularly good bound using a different method.

Generalizing the concept of a perfect number is a Zumkeller or integer perfect number that was introduced by Zumkeller in 2003. The positive integer $n$ is a Zumkeller number if its divisors can be partitioned into two sets with the same sum, which will be $σ(n)/2$. Generalizing even further, we call $n$ a $k$-layered number if its divisors can be partitioned into $k$ sets with equal sum. In this paper, we completely characterize Zumkeller numbers with two distinct prime factors and give some bounds for prime factorization in case of Zumkeller numbers with more than two distinct prime factors. We also characterize $k$-layered numbers with two distinct prime factors and even $k$-layered numbers with more than two distinct odd prime factors. Some other results concerning these numbers and their relationship with practical numbers and Harmonic mean numbers are also discussed.

Using the Wolfram NumberTheory package and the Recognize command, together with numerical estimates involving the elliptic lambda and elliptic alpha functions, Bagis and Glasser, in 2013, introduced a conjectural Ramanujan-type series related to the class number $h(-d) = 1$ for a quadratic form with discriminant $d = 163$. This conjectured series is of level one and has positive terms, and recalls the Chudnovsky brothers' alternating series of the same level, given the connection between the Chudnovsky-Chudnovsky formula and the Heegner number $d = 163$ such that $\mathbb{Q}\left( \sqrt{-d} \right)$ has class number one. We prove Bagis and Glasser's conjecture by proving evaluations for $λ^{\ast}(163)$ and $α(163)$, which we derive using the Chudnovsky brothers' formula together with the analytic continuation of a formula due to the Borwein brothers for Ramanujan-type series of level one. As a byproduct of our method, we obtain an infinite family of Ramanujan-type series for $\frac{1}π$ generalizing the Chudnovsky algorithm.

Let $f$ be a newform of weight $k\geq 2$, level $N$ with coefficients in a number field $K$, and $A$ the adjoint motive of the motive $M$ associated to $f$. We carefully discuss the construction of the realisations of $M$ and $A$, as well as natural integral structures in these realisations. We then use the method of Taylor and Wiles to verify the $λ$-part of the Tamagawa number conjecture of Bloch and Kato for $L(A,0)$ and $L(A,1)$. Here $λ$ is any prime of $K$ not dividing $Nk!$, and so that the mod $λ$ representation associated to $f$ is absolutely irreducible when restricted to the Galois group over $\mathbb{Q}(\sqrt{(-1)^{(\ell-1)/2}\ell})$ where $λ\mid \ell$. The method also establishes modularity of all lifts of the mod $λ$ representation which are crystalline of Hodge-Tate type $(0,k-1)$.

We give new examples of pairs composed of a real and a $p$-adic numbers that satisfy a conjecture on simultaneous multiplicative approximation by rational numbers formulated by Einsiedler and Kleinbock in 2007.

Let $A$ be a non-CM simple abelian variety over a number field $K$. For a place $v$ of $K$ such that $A$ has good reduction at $v$, let $F(A,v)$ denote the Frobenius field generated by the corresponding Frobenius eigenvalues. Assuming $A$ has connected monodromy groups, we show that the set of places $v$ such that $F(A,v)$ is isomorphic to a fixed number field has upper Dirichlet density zero. Assuming the GRH, we give a power saving upper bound for the number of such places.

Using Galois representations, we analyze fields of definition of cyclic isogenies on elliptic curves to prove the following uniformity result: for any number field $F$ which has no rational CM, under GRH there exists an effectively computable constant $B:=B(F)\in\mathbb{Z}^+$ such that for any finite extension $L/F$ whose degree $[L:F]$ is coprime to $B$, one has for all elliptic curves $E_{/F}$ that any $L$-rational isogeny on $E$ is $F$-rational. For any number field $F$, under GRH we also prove results for the mod-$\ell$ Galois representations of non-CM elliptic curves with an $F$-rational isogeny of uniformly large prime degree $\ell$.

In this article, we study the asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields $K$, i.e., $Ax^p+By^p=Cz^3$ with prime exponent $p$ and $A,B,C \in \mathcal{O}_K \setminus \{0\}$. For certain class of fields $K$, we prove that $Ax^p+By^p=Cz^3$ has no asymptotic solutions over $K$ (resp., solutions of certain type over $K$) with restrictions on $A,B,C$ (resp., for all $A,B,C \in \mathcal{O}_K \setminus \{0\}$). Finally, we present several local criteria over $K$.

Let $K$ be a number field of degree $d$. Then every ideal $I$ in the ring of integers ${\mathcal O}_K$ contains infinitely many primitive elements, i.e. elements of degree $d$. A bound on smallest height of such an element in $I$ follows from some recent developments in the direction of a 1998 conjecture of W. Ruppert. We prove a very explicit bound like this in the case of quadratic fields. Further, we consider primitive elements in an ideal outside of a finite union of other ideals and prove a bound on the height of a smallest such element. Our main tool is a result on points of small norm in a lattice outside of an algebraic hypersurface and a finite union of sublattices of finite index, which we prove by blending two previous Diophantine avoidance results. We also obtain an avoidance result like this for lattice points in the positive orthant in $\mathbb{R}^d$ and use it to obtain a small-height totally positive primitive element in an ideal of a totally real number field outside of a finite union of other ideals. Additionally, we use our avoidance method to prove a bound on the Mahler measure of a generating non-sparse polynomial for a given number field. Finally, we produce a

Given a number field $L$, we define the degree of an algebraic number $v \in L$ with respect to a choice of a primitive element of $L$. We propose the question of computing the minimal degrees of algebraic numbers in $L$, and examine these values in degree $4$ Galois extensions over $\mathbb{Q}$ and triquadratic number fields. We show that computing minimal degrees of non-rational elements in triquadratic number fields is closely related to solving classical Diophantine problems such as congruent number problem as well as understanding various arithmetic properties of elliptic curves.

We investigate the $a$-numbers of $\mathbb{Z}/p^2\mathbb{Z}$-covers in characteristic $p>2$ and extend a technique originally introduced by Farnell and Pries for $\mathbb{Z}/p\mathbb{Z}$-covers. As an application of our approach, we demonstrate that the $a$-numbers of ``minimal'' $\mathbb{Z}/9\mathbb{Z}$-covers can be deduced from the associated branching datum.

The goal of this note is to provide a general lower bound on the number of even values of the Fourier coefficients of an arbitrary eta-quotient $F$, over any arithmetic progression. Namely, if $g_{a,b}(x)$ denotes the number of even coefficients of $F$ in degrees $n\equiv b$ (mod $a$) such that $n\le x$, then we show that $g_{a,b}(x) / \sqrt{x}$ is unbounded for $x$ large. Note that our result is very close to the best bound currently known even in the special case of the partition function $p(n)$ (namely, $\sqrt{x}\log \log x$, proven by Bellaïche and Nicolas in 2016). Our argument substantially relies upon, and generalizes, Serre's classical theorem on the number of even values of $p(n)$, combined with a recent modular-form result by Cotron \emph{et al.} on the lacunarity modulo 2 of certain eta-quotients. Interestingly, even in the case of $p(n)$ first shown by Serre, no elementary proof is known of this bound. At the end, we propose an elegant problem on quadratic representations, whose solution would finally yield a modular form-free proof of Serre's theorem.

Let $π(x;q,a)$ denote the number of primes up to $x$ that are congruent to $a$ modulo $q$. A prime number race, for fixed modulus $q$ and residue classes $a_1, \ldots, a_r$, investigates the system of inequalities $π(x;q,a_1) > π(x;q,a_2) > \cdots > π(x;q,a_r)$. The study of prime number races was initiated by Chebyshev and further studied by many others, including Littlewood, Shanks-Rényi, Knapowski-Turan, and Kaczorowski. We expect that this system of inequalities should have arbitrarily large solutions $x$, and moreover we expect the same to be true no matter how we permute the residue classes $a_j$; if this is the case, and if the logarithmic density of the set of such $x$ exists and is positive, the prime number race is called inclusive. In breakthrough research, Rubinstein and Sarnak proved conditionally that every prime number race is inclusive; they assumed not only the generalized Riemann hypothesis but also a strong statement about the linear independence of the zeros of Dirichlet $L$-functions. We show that the same conclusion can be reached assuming the generalized Riemann hypothesis and a substantially weaker linear independence hypothesis. In fact, we can ass

We use representations and differentiation algorithms of posets, in order to obtain results concerning unsolved problems on figurate numbers. In particular, we present criteria for natural numbers which are the sum of three octahedral numbers, three polygonal numbers of positive rank or four cubes with two of them equal. Some identities of the Rogers-Ramanujan type involving this class of numbers are also obtained.

Let $\ell$ and $p$ be two distinct primes. We study the $p$-adic valuation of the number of spanning trees in an abelian $\ell$-tower of connected multigraphs. This is analogous to the classical theorem of Washington--Sinnott on the growth of the $p$-part of the class group in a cyclotomic $\mathbb{Z}_\ell$-extension of abelian extensions of $\mathbb{Q}$. Furthermore, we show that under certain hypotheses, the number of primes dividing the number of spanning trees is unbounded in such a tower.

Let $d\geq 1$ be an integer and let $p$ be a rational prime. Recall that $p$ is a torsion prime of degree $d$ if there exists an elliptic curve $E$ over a degree $d$ number field $K$ such that $E$ has a $K$-rational point of order $p$. Derickx, Kamienny, Stein and Stoll have computed the torsion primes of degrees 4, 5, 6 and 7; we verify that these techniques can be extended to determine the torsion primes of degree 8.

Frobenius problem and its many generalizations have been extensively studied in several areas of mathematics. We study semigroups of totally positive algebraic integers in totally real number fields, defining analogues of the Frobenius numbers in this context. We use a geometric framework recently introduced by Aliev, De Loera and Louveaux to produce upper bounds on these Frobenius numbers in terms of a certain height function. We discuss some properties of this function, relating it to absolute Weil height and obtaining a lower bound in the spirit of Lehmer's conjecture for algebraic vectors satisfying some special conditions. We also use a result of Borosh and Treybig to obtain bounds on the size of representations and number of elements of bounded height in such positive semigroups of totally real algebraic integers.

Given $k, \ell \in {\bf N}^+$, let $x_{i,j}$ be, for $1 \le i \le k$ and $0 \le j \le \ell$, some fixed integers, and define, for every $n \in {\bf N}^+$, $s_n := \sum_{i=1}^k \prod_{j=0}^\ell x_{i,j}^{n^j}$. We prove that the following are equivalent: (a) There are a real $θ> 1$ and infinitely many indices $n$ for which the number of distinct prime factors of $s_n$ is greater than the super-logarithm of $n$ to base $θ$. (b) There do not exist non-zero integers $a_0,b_0,\ldots,a_\ell,b_\ell $ such that $s_{2n}=\prod_{i=0}^\ell a_i^{(2n)^i}$ and $s_{2n-1}=\prod_{i=0}^\ell b_i^{(2n-1)^i}$ for all $n$. We will give two different proofs of this result, one based on a theorem of Evertse (yielding, for a fixed finite set of primes $\mathcal S$, an effective bound on the number of non-degenerate solutions of an $\mathcal S$-unit equation in $k$ variables over the rationals) and the other using only elementary methods. As a corollary, we find that, for fixed $c_1, x_1, \ldots,c_k, x_k \in \mathbf N^+$, the number of distinct prime factors of $c_1 x_1^n+\cdots+c_k x_k^n$ is bounded, as $n$ ranges over $\mathbf N^+$, if and only if $x_1=\cdots=x_k$.