For a set A of nonnegative integers, let R2(A,n) denote the number of solutions to n=a+a′ with a,a′∈A, a

Let d be a positive square-free integer and 𝜖d=(Td+Udd)/2>1 be the fundamental unit of the real quadratic field ℚ(d). The Ankeny–Artin–Chowla (AAC) conjecture asserts that Up≢0(modp) for primes p≡1(mod4), which still remains unsolved. In this paper, sufficient conditions for Ud

Given a number field L, we define the degree of an algebraic number v∈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 ℚ and triquadratic number fields. We show that computing minimal degrees of non-rational elements in triquadratic number fields is

We generalize our work on Carlitz prime power torsion extension to torsion extensions of Drinfeld modules of arbitrary rank. As in the Carlitz case, we give a description of these extensions in terms of evaluations of Anderson generating functions and their hyperderivatives at roots of unity. We also give a direct proof that the image of the Galois representation attached to the 𝔭-adic Tate module

In this paper, we study quadratic points on the non-split Cartan modular curves Xns(p), for p=7,11, and 13. Recently, Siksek proved that all quadratic points on Xns(7) arise as pullbacks of rational points on Xns+(7). Using similar techniques for p=11, and employing a version of Chabauty for symmetric powers of curves for p=13, we show that the same holds for Xns(11) and Xns(13). As a consequence,

Let K=𝔽p(z1,…,zr) be a finitely generated field over 𝔽p and fix a,b∈K∗. We study the solutions of the Catalan equation axm+byn=1 to be solved in x,y∈K and integers m,n>1 coprime with p. Our main result corrects earlier work of Silverman.

In this paper, we give an explicit upper bound on h(p), the least primitive root modulo p2. Since a primitive root modulo p is not primitive modulo p2 if and only if it belongs to the set of integers less than p which are pth power residues modulo p2, we seek the bounds for N1(H) and N2(H) to find H which satisfies N1(H)−N2(H)>0, where, N1(H) denotes the number of primitive roots modulo p not exceeding

Let b≥2 be an integer, and write the base b expansion of any non-negative integer n as n=x0+x1b+⋯+xdbd, with xd>0 and 0≤xi0 for all n>0. Consider the map Sϕ,b:ℤ≥0→ℤ≥0, with Sϕ,b(n):=ϕ(x0)+⋯+ϕ(xd). It is known that the orbit set {n,Sϕ,b(n),Sϕ,b(Sϕ,b(n)),…,} is finite for all n>0. Each orbit contains a finite cycle, and for a given b, the union of such cycles over all orbit sets is finite. Fix now an

Let ck,j(n) be the number of (k,j)-colored partitions of n. Recently, Keith proved that for j∈{4,7}, if c9,j(3ln+2)≡0(mod27) for all n≥0, then l is large. We prove that such l do not exist. Furthermore, for any positive integers a,b with (a,b)=1, there exist infinitely many positive integers n such that c9,j(an+b)≢0(mod27), where j∈{4,7}.

Let K=ℚ(α) be a pure number field generated by a complex root α of a monic irreducible polynomial F(x)=xpr−m where m≠1 is a square free rational integer, p is a rational prime integer, and r is a positive integer. In this paper, we study the monogenity of K. We prove that if νp(mp−m)=1, then K is monogenic. But if r≥p and νp(mp−m)>p, then K is not monogenic. Some illustrating examples are given.

Recently the first author proved a congruence proposed in 2006 by Adamchuk: ∑k=1⌊2p/3⌋2kk≡0(modp2) for any prime p=1(mod3). In this paper, we provide more examples (with proofs) of congruences of the same kind ∑k=1⌊ap/r⌋2kkxk(modp2) where p is a prime such that p≡1(modr), a/r is a fraction in (1/2,1) and x is a p-adic integer. The key ingredients are the p-adic Gamma function Γp and a special class

We introduce a certain multiple Hurwitz zeta function as a generalization of the Mordell–Tornheim multiple zeta function, and study its analytic properties. In particular, we evaluate the values of the function and its first and second derivatives at non-positive integers.

1 It is a subtle question as to when the Diophantine equation of the tittle has solutions in positive integers. Here, we show that the equation in the title does not have solutions in positive integers in the case that n is of the form n=4q, where q2−1=2hq1, with h,q1∈ℤ+, 2|h, h≥4, and 8|q1+1. We do this by explicitly calculating a Brauer–Manin obstruction to weak approximation on the elliptic surface

Any integral convex polytope P in ℝN provides an N-dimensional toric variety XP and an ample divisor DP on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on XP, obtained by evaluating global section of the line bundle corresponding to DP on every rational point of XP. This work presents an extension of toric codes analogous to the one of Reed–Muller

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal 𝔩 of 𝔽q[T], the question essentially asks whether, up to isogeny, a Drinfeld module ϕ over 𝔽q(T) contains a rational 𝔩-torsion point if the reduction of ϕ at almost all primes of 𝔽q[T] contains

In this paper, we establish a strategy for the calculation of the proportion of everywhere locally soluble diagonal hypersurfaces of ℙn of fixed degree. Our strategy is based on the product formula established by Bright, Browning and Loughran. Their formula reduces the problem into the calculation of the proportions of ℚv-soluble diagonal hypersurfaces for all places v. As worked examples, we carry

Let Λ(n) be the von Mangoldt function, let n≥2 be an integer and let RG(n;q,a,b):=∑m1+m2=nm1≡amodqm2≡bmodqΛ(m1)Λ(m2) be the counting function for the Goldbach numbers with summands in arithmetic progression modulo a common integer q. We prove an asymptotic formula for the weighted average, with Cesàro weight of order k>1, with k∈ℝ, of this function. Our result is uniform in a suitable range for q.

Let χ be a real and non-principal Dirichlet character, L(s,χ) its Dirichlet L-function and let p be a generic prime number. We prove the following result: If for some 0≤σ<1 the partial sums ∑p≤xχ(p)p−σ change sign only for a finite number of integers x, then there exists 𝜖>0 such that L(s,χ) has no zeros in the half plane Re(s)>1−𝜖.

Let F and G be Siegel cusp forms for the group Sp4(ℤ) with weights k1, k2, respectively. Suppose that neither F nor G is a Saito–Kurokawa lift. Further suppose that F and G are Hecke eigenforms lying in distinct eigenspaces. In this paper, we investigate simultaneous arithmetic behavior and related problems of Hecke eigenvalues of these Hecke eigenforms, some of which improve upon results of Gun et al

In this paper, for a geometrically integral projective scheme, we will give an upper bound of the product of the norms of its non-geometrically integral reductions over an arbitrary number field. For this aim, we take the adelic viewpoint on this subject. Résumé (Contrôle des réductions non-géométriquement intègres) Dans cet article, pour un schéma projectif géométriquement intègre, on donnera une

In M. Ru and P. Vojta, A birational Nevanlinna constant and its consequences, Amer. J. Math.142(3) (2020) 957–991, among other things, proved the so-called general theorem (arithmetic part) which can be viewed as an extension of Schmidt’s subspace theorem. In this paper, we extend their result by replacing the divisors by closed subschemes.

The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that ρ̄:Gal(ℚ̄/ℚ)→GL2(𝔽̄p) lifts to a Hida line for which the weights range over a congruence class modulo-p2. The advantage of the purely Galois theoretic approach is that it allows us to construct p-adic families of Galois representations lifting the actual representation

Let NT(r,k,n) count the total number of parts among partitions of n with rank congruent to r modulo k and let Mω(r,k,n) count the total appearances of ones among partitions of n with crank congruent to r modulo k. We provide a list of over 70 congruences modulo 5, 7, 11 and 13 involving NT(r,k,n) and Mω(r,k,n), which are known as congruences of Andrews–Beck type. Some recent conjectures of Chan, Mao

A finite set of prime numbers S is called unavoidable with respect to b>1 if for each ξ∈ℝ the sequence of integer parts ⌊ξbn⌋, n=0,1,2,…, contains infinitely many elements divisible by at least one prime number p from the set S. It is known that an unavoidable set exists with respect to b=2,3,4,6 and that it does not exist if b>1 is an integer such that b−1 is not square free. In this paper, we show

Let H be a positive integer and let p be an odd prime. For 1≤x,y≤H, we study the distribution of consecutive square-free numbers of the forms x2+y2+1, x2+y2+2 and xy+1, xy+2. In addition, we study the distribution of consecutive square-free primitive roots modulo p of the forms x2+y2+1,x2+y2+2, xy+1, xy+2, and x2+1, x2+2, respectively.

In this paper, we give linear relations between the Fourier coefficients of a special Hilbert modular form of half integral weight and some arithmetic functions. As a result, we have linear relations for the special L-values over certain totally real number fields.

We first study the mean value of certain restricted divisor sums involving the Chowla–Walum sums, improving in particular a recent estimate given by Iannucci. The aim of the second part of this work is the generalization of the previous study, by restricting the range of the divisors in the studied divisor sums, extending the Chowla–Walum conjecture, proving a small part of this extended conjecture

In this paper, we prove that there are no positive integers a, b, c and d such that {Pa,Pb,Pc,Pd} is a Diophantine quadruple, where for a positive integer m, Pm is the mth Pell number.

A positive integer n is called weakly prime-additive if n has at least two distinct prime divisors and there exist distinct prime divisors p1,…,pt of n and positive integers α1,…,αt such that n=p1α1+⋯+ptαt. It is easy to see that t≥3. In this paper, intrigued by De Koninck and Luca’s work, we further determine all weakly prime-additive numbers n such that n=22+p2+q2, where p,q are distinct odd prime

We show that if V and V′ are two p-adic representations of Gal(Q¯p/Qp) whose tensor product is trianguline, then V and V′ are both potentially trianguline.

In this paper, we show that if (Xn,Yn) is the nth solution of the Pell equation X2−dY2=±1 for some non-square d, then given any integer c, the equation c=Xn−2m has at most 2 integer solutions (n,m) with n≥0 and m≥0, except for the only pair (c,d)=(−1,2). Moreover, we show that this bound is optimal. Additionally, we propose a conjecture about the number of solutions of Pillai’s problem in linear recurrent

Finding all integers which can be written as a sum of three nonzero squares of integers has been studied by a number of authors. This question is solved under the assumption of the Generalized Riemann Hypothesis (GRH), but still remains unsolved unconditionally. In this paper, we show that out of all integers that are sums of three squares, all but finitely many can be written as x2+y2+z2 for some

We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors. First, given an odd prime q and a finite abelian q-group H, we consider the set of integers n≤x such that the Sylow q-subgroup of the multiplicative group (ℤ/nℤ)× is isomorphic to H. We show that the counting function of this

We study some classical identities for multiple zeta values and show that they still hold for zeta functions built from an arbitrary sequence of nonzero complex numbers. We introduce the complementary zeta function of a system, which naturally occurs when lifting identities for multiple zeta values to identities for quasisymmetric functions.

Let τ(n) be Ramanujan’s tau function, defined by the discriminant modular form Δ(z)=q∏j=1∞(1−qj)24=∑n=1∞τ(n)qn,q=e2πiz (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer’s conjecture asserts that τ(n)≠0 for all n≥1; since τ(n) is multiplicative, it suffices to study primes p for which τ(p) might possibly be zero. Assuming standard conjectures for the twisted

Let N/K be a Galois extension of p-adic number fields and let V be a de Rham representation of the absolute Galois group GK of K. In the case V=ℚp(1), the equivariant local 𝜖-constant conjecture describes the compatibility of the equivariant Tamagawa number conjecture with the functional equation of Artin L-functions and it can be formulated as the vanishing of a certain element RN/K in K0(ℤp[G],ℚpc[G]);

Let f:ℤ+→ℤ+ be a polynomial with the property that corresponding to every prime p there exists an integer ℓ such that p∤f(ℓ). In this paper, we establish some equidistributed results between the number of partitions of an integer n whose parts are taken from the sequence {f(ℓ)}ℓ=1∞ and the number of parts of those partitions which are in a certain arithmetic progression.

We consider the action of Hecke-type operators on Hilbert–Siegel theta series attached to lattices of even rank. We show that the average Hilbert–Siegel theta series are eigenforms for these operators, and we explicitly compute the eigenvalues.

Let a,b be fixed positive integers such that a is not a perfect square and b is squarefree, and let ω(b) denote the number of distinct prime divisors of b. Let (u1, v1) denote the least solution of Pell equation u2−av2=1. Further, for any positive integer n, let un=αn+ᾱn2 and vn=αn−ᾱn2a, where α=u1+v1a and ᾱ=u1−v1a. In this paper, using the basic properties of Pell equations and some known results

Let (Fn)n≥1 be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that loglcm(F1,F2,…,Fn)∼3logαπ2⋅n2,as n→+∞, where lcm is the least common multiple and α:=1+5)/2 is the golden ratio. We prove that for every periodic sequence s=(sn)n≥1 in {−1,+1} there exists an effectively computable rational number Cs>0 such that loglcm(F3+s3,F4+s4,…,Fn+sn)∼3logαπ2⋅Cs⋅n2,as n→+∞. Moreover, we show that

This paper describes cubic function fields L/K with prescribed ramification, where K is a rational function field. We give general equations for such extensions, an explicit procedure to obtain a defining equation when the purely cubic closure K′/K of L/K is of genus zero, and a description of the twists of L/K up to isomorphism over K. For cubic function fields of genus at most one, we also describe

In this paper, we consider the Diophantine equation in the title, where p,q are distinct odd prime numbers and a,b are natural numbers. We present many results given conditions for the existence of integers solutions for this equation, according to the values of p,q,a and b. Our methods are elementary in nature and are based upon the study of the primitive divisors of certain Lucas sequences as well

The goal of this paper is to define an analogue of the Weil-pairing for Drinfeld modules using explicit formulas and to deduce its main properties from these formulas. Our result generalizes the formula given for rank 2 Drinfeld modules by van der Heiden and works as a more explicit, elementary proof of the Weil-pairing’s existence given by van der Heiden.

An l-regular overpartition of n is an overpartition of n into parts not divisible by l. Let A¯l(n) be the number of l-regular overpartitions of n. Andrews defined singular overpartitions counted by the partition function C¯k,i(n). It denotes the number of overpartitions of n in which no part is divisible by k and only parts≡±i(modk) may be overlined. He proved that C¯3,1(9n+3) and C¯3,1(9n+6) are divisible

We consider the identity component of the Sato–Tate group of the Jacobian of curves of the form C1:y2=x2g+2+c,C2:y2=x2g+1+cx,C3:y2=x2g+1+c, where g is the genus of the curve and c∈ℚ∗ is constant. We approach this problem in three ways. First we use a theorem of Kani-Rosen to determine the splitting of Jacobians for C1 curves of genus 4 and 5 and prove what the identity component of the Sato–Tate group

We prove an explicit version of Burgess’ bound on character sums for composite moduli.

In this note, we prove a conjecture of Reitzes, Vulakh and Young on the location of the zeros of certain cuspforms of level one.

In this paper, we obtain an upper bound for the number of integral solutions, of given height, of system of two quadratic forms in five variables. Our bound is an improvement over the bound given in [H. Iwaniec and R. Munshi, The circle method and pairs of quadratic forms, J. Théor. Nr. Bordx.22 (2010) 403–419].

For any positive integer k≥1, let p1/k(n) be the number of solutions of the equation n=[a1k]+⋯+[adk] with integers a1≥⋯≥ad≥1, where [t] is the integral part of real number t. Recently, Luca and Ralaivaosaona gave an asymptotic formula for p1/2(n). In this paper, we give an asymptotic development of p1/k(n) for all k≥1. Moreover, we prove that the number of such partitions is even (respectively, odd)

We prove evaluations of Hankel determinants of linear combinations of moments of orthogonal polynomials (or, equivalently, of generating functions for Motzkin paths), thus generalizing known results for Catalan numbers.

Congruences and related identities are derived for a set of colored and weighted partition functions whose generating functions generate the graded algebra of integer weight modular forms of level seven. The work determines a general strategy for identifying and proving identities and associated congruences for modular forms on the principal congruence subgroup of level 7. Ramanujan’s partition congruence

Let f(n) and g(n) be the number of unordered and ordered factorizations of n into integers larger than one. Let F(n) and G(n) have the additional restriction that the factors are coprime. We establish asymptotic bounds for the sums of F(n)β and G(n)β up to x for all real β and the asymptotic bounds for f(n)β and g(n)β for all negative β.

We investigate the order of exponential sums involving the coefficients of general L-functions satisfying a suitable functional equation and give some new estimates, including refining certain results in preceding works [X. Ren and Y. Ye, Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for GLm(ℤ), Sci. China Math.58(10) (2015) 2105–2124; Y. Jiang and G. Lü, Oscillations

Let k be a number field, Ok its ring of integers, Cl(k) its classgroup and h the class number of k. Let Γ be a finite group. Let ℳ be a maximal Ok-order in the semi-simple algebra k[Γ] containing Ok[Γ], and Cl(ℳ) its locally free classgroup. Let A=ℤ∪{1m,m∈ℤ∖{0,1,−1}} and n∈A. We define the set ℛ(𝒟n,ℳ) of Galois module classes realizable by the nth power of the different to be the set of classes c∈Cl(ℳ)

In this paper we prove that for every integer d≥1, there exists an explicit constant Bd such that the following holds. Let K be a number field of degree d, let q>max{d−1,5} be any rational prime that is totally inert in K and E any elliptic curve defined over K such that E has potentially multiplicative reduction at the prime 𝔮 above q. Then for every rational prime p>Bd, E has an irreducible mod

Let d be an odd positive square-free integer. In this paper, we shall investigate the structure of the 2-class group of the cyclotomic ℤ2-extension of the imaginary biquadratic number field ℚ(d,−1) if d is of specifiic form. Furthermore, we deduce the structure of the 2-class group of the cyclotomic ℤ2-extension of ℚ(−d).

Given a weight vector τ=(τ1,…,τn)∈ℝ+n with each τi bounded by certain constraints, we obtain a lower bound for the Hausdorff dimension of the set 𝒲n(τ)∩ℳ, where ℳ is a twice continuously differentiable manifold. From this we produce a lower bound for 𝒲n(Ψ)∩ℳ where Ψ is a general approximation function with certain limits. The proof is based on a technique developed by Beresnevich et al. in 2017,

For each algebraic number α∈ℚ̄, a result of Habegger [P. Habegger, Singular moduli that are algebraic units, Algebra Number Theory 9(7) (2015) 1515–1524] shows that there are only finitely many singular moduli j such that j−α is an algebraic unit. His result uses Duke’s Equidistribution Theorem and is thus not effective. In this paper, we give an effective proof of Habegger’s result assuming that α

We give two variations on a result of Wilkie’s [A. J. Wilkie, Complex continuations of ℝan,exp-definable unary functions with a diophantine application, J. Lond. Math. Soc. (2) 93(3) (2016) 547–566] on unary functions definable in ℝan,exp that take integer values at positive integers. Provided that the function grows slower (in a suitable sense) than the function 2x, Wilkie showed that it must be eventually

For an algebraic number field K, let ζK(s) be the associated Dedekind zeta-function. It is conjectured that ζK(m) is transcendental for any positive integer m>1. The only known case of this conjecture was proved independently by Siegel and Klingen, namely that, when K is a totally real number field, ζK(2n) is an algebraic multiple of π2n[K:ℚ] and hence, is transcendental. If K is not totally real,