International Journal of Number Theory

A well-known conjecture asserts that there are infinitely many primes p for which p - 1 is a perfect square. We obtain upper and lower bounds of matching order on the number of pairs of distinct primes p, q ≤ x for which (p - 1)(q - 1) is a perfect square.

We prove an orthogonality relation for the Fourier–Whittaker coefficients of a thin family of GL(3) Maass forms containing all self-dual forms. This is obtained by analyzing the Kuznetsov trace formula on GL(3) for a certain family of test functions. The method also yields Weyl's law for the same family of Maass forms.

Let B be a set of natural numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B = {bb′|b, b′ ∈ B} cannot be greater than O(n log n) which matches the lower bound provided in an earlier paper up to a multiplicative constant. For sets of complex numbers, we improve the bound to O_ϵ(n^{1 + ϵ}) for arbitrary ϵ > 0 assuming the GRH.

This paper studies a formal power series with coefficients in a Hecke ring associated with the Heisenberg Lie algebra. We relate the series to the classical Hecke series defined by Hecke, and prove that the series has a property similar to the rationality theorem of the classical Hecke series. And then, our results recover the rationality theorem of the classical Hecke series.

We introduce the inversion polynomial for Dedekind sums f_b(x) = ∑ x^{inv(a, b)} to study the number of s(a, b) which have the same value for a given b. We prove several properties of this polynomial and present some conjectures. We also introduce connections between Kloosterman sums and the inversion polynomial evaluated at particular roots of unity. Finally, we improve on previously known bounds for the second highest value of the Dedekind sum and provide a conjecture for a possible generalization. Lastly, we include a new sufficient condition for the inequality of two Dedekind sums based on the reciprocity formula.

Soient K un corps de nombres et p un nombre premier ≥ 5. Notons μ_p le groupe des racines p-ièmes de l'unité. On définit p comme étant K-régulier si p ne divise pas le nombre de classes du corps K(μ_p). Sous l'hypothèse que p est K-régulier et inerte dans K, on établit le second cas du théorème de Fermat sur K pour l'exposant p. On utilise pour cela des arguments classiques, ainsi que le théorème de Faltings selon lequel une courbe de genre au moins deux sur K n'a qu'un nombre fini de points K-rationnels. De plus, si K est un corps quadratique imaginaire, distinct de , on en déduit un énoncé permettant souvent en pratique de démontrer le théorème de Fermat sur K pour un exposant K-régulier donné.

Mots-clés: Théorème de Fermat; corps de nombres.

Let K be a number field and p a prime number ≥5. Let us denote by μ_p the group of the pth roots of unity. We define p to be K-regular if p does not divide the class number of the field K(μ_p). Under the assumption that p is K-regular and inert in K, we establish the second case of Fermat's Last Theorem over K for the exponent p. We use in the proof classical arguments, as well as Faltings' theorem stating that a curve of genus at least two over K has a finite number of K-rational points. Moreover, if K is an imaginary quadratic field, other than , we deduce a statement which allows often in practice to prove Fermat's Last Theorem over K for a given K-regular exponent.

Using modular forms, we determine the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients 1, 3 or 9.

In this paper, we study some exceptional sets of points whose partial quotients in their Sylvester continued fraction expansions obey some restrictions. More precisely, for α ≥ 1 we prove that the Hausdorff dimension of the set

We determine the coefficients of the Fourier series of a class of eta quotients of weight 2. For example, we show that

Let p > 3 be a prime, and let a be a rational p-adic integer. Let {B_n(x)} denote the Bernoulli polynomials given by B₀ = 1, and . In this paper, using Bernoulli polynomials we establish congruences for and . As a consequence we solve the following conjecture of Z. W. Sun:

The purpose of this paper is to give an explicit description, in terms of hypergeometric functions over finite fields, of the zeta functions of certain smooth hypersurfaces that generalize the Dwork family. The key point here is that we count the number of rational points employing both the techniques of character sums and the theory of weights, which enables us to enlighten the calculation of the zeta function.

Although links between values of finite field hypergeometric functions and eigenvalues of elliptic modular forms are well known, we establish in this paper that there are also connections to eigenvalues of Siegel modular forms of higher degree. Specifically, we relate the eigenvalue of the Hecke operator of index p of a Siegel eigenform of degree 2 and level 8 to a special value of a ₄F₃-hypergeometric function.

In 1940, Weil [Sur les fonctions algébriques à corps de constantes finis, C. R. Acad. Sci. Paris210 (1940) 592–594] proved the Riemann hypothesis for curves over finite fields. It follows from the Castelnuovo–Severi defect inequality concerning correspondences between algebraic curves (see [A. Mattuck and J. Tate, On the inequality of Castelnuovo–Severi, Abh. Math. Sem. Univ. Hamburg22 (1958) 295–299]). An important step in the proof of Castelnuovo–Severi's defect inequality is the invariance of the Castelnuovo–Severi defect under trivial correspondences, so that the degree of divisors can be modified by adding multiples of trivial correspondences. In the number field case, the Weil distribution Δ(h) (see [A. Weil, Sur les formules explicites de la théorie des nombres, Izv. Akad. Nauk SSSR Ser. Mat.36 (1972) 3–18]) corresponds to the Castelnuovo–Severi defect. Functions of the form ∑_ξ∈K* f(ξx) with f in the Schwartz–Bruhat space S(𝔸)₀ correspond to trivial correspondences. In this paper, we show that the two terms and in the Weil distribution can be chosen to be zero by adding "trivial correspondences" to h while keeping the Weil distribution essentially unchanged. As an application of this result, the Weil distribution is expressed as the spectral trace of an operator on a Hilbert space.

We prove a general theorem that evaluates the Legendre symbol under certain conditions on the integers A, B, m and the prime p. The evaluation is in terms of parameters appearing in a binary quadratic form representing p. The theorem has applications to quartic residuacity.

We prove that the greatest positive integer that is not expressible as a linear combination with integer coefficients of elements of the set {n², (n + 1)², …} is asymptotically O(n²), verifying thus a conjecture of Dutch and Rickett. Furthermore, we ask a question on the representation of integers as sum of four large squares.