International Journal of Number Theory

In this paper, we propose different notions of 𝔽_ζ-geometry, for ζ a root of unity, generalizing notions of 𝔽₁-geometry (geometry over the "field with one element") based on the behavior of the counting functions of points over finite fields, the Grothendieck class, and the notion of torification. We relate 𝔽_ζ-geometry to formal roots of Tate motives, and to functions in the Habiro ring, seen as counting functions of certain ind-varieties. We investigate the existence of 𝔽_ζ-structures in examples arising from general linear groups, matrix equations over finite fields, and some quantum modular forms.

We prove an equidistribution result for the Satake parameters of the local representations attached to Siegel cusp forms of degree 2 of increasing level and weight, counted with a certain arithmetic weight. We then apply this to compute the symmetry type of a similarly weighted distribution of the low-lying zeros of L-functions attached to these cusp forms.

Let k be a global field. Let G be a connected linear algebraic k-group, assumed reductive when k is a function field. It follows from a result of a paper by Bary-Soroker, Fehm and Petersen that when H is a smooth connected k-subgroup of G, the quotient space G/H is of Hilbert type. We prove a similar result for certain non-connected k-subgroups H of G. In particular, we prove that if G is a simply connected k-group over a number field k, and H is an abelian k-subgroup of G, not necessarily connected, then G/H is of Hilbert type.

It is well known that all numbers that are normal of order k in base b are also normal of all orders less than k. Another basic fact is that every real number is normal in base b if and only if it is simply normal in base b^k for all k. This may be interpreted to mean that a number is normal in base b if and only if all blocks of digits occur with the desired relative frequency along every infinite arithmetic progression. We reinterpret these theorems for the Q-Cantor series expansions and show that they are no longer true in a particularly strong way. The main theoretical result of this paper will be to reduce the problem of constructing normal numbers with certain pathological properties to the problem of solving a system of Diophantine relations.

In this paper, we get several new results on permutation polynomials over finite fields. First, by using the linear translator, we construct permutation polynomials of the forms and . These forms generalize the results obtained by Kyureghyan in 2011. Consequently, we characterize permutation polynomials of the form , which extends a theorem of Charpin and Kyureghyan obtained in 2009.

A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a metrical and a non-metrical point of view and, on the other hand, from an asymptotic and also a uniform point of view. The principal novelty is a Khintchine type theorem for uniform approximation in this context. Some applications of this theory are also discussed.

We shall obtain the irrationality exponent 2 for some values of two special Mahler functions. This gives a new proof for the recent result of Bugeaud on Thue–Morse–Mahler numbers.

Let a = {a₁,…, a_k} be a finite multiset of positive integers. Consider the sequence of all positive integer multiples of all a_is, and note the multiplicity of each term in this sequence. This sequence of multiplicities is called the resonance sequence generated by {a₁,…, a_k}. Two multisets are called combinatorially equivalent if they generate the same resonance sequence. Sometimes the initial set can be recovered from the resonance sequence, if we assume some additional information. These sets are called recoverable. The paper gives some criteria of recoverability for a rather generic class of generating sets.

Using two expansion formulas for the Rogers–Szegő polynomials and the Stieltjes–Wigert polynomials, we give new proofs of a variety of important classical formulas including Bailey's ₆ψ₆ summation, the Askey–Wilson integral and its extension. Furthermore, we give nontrivial extensions of the Andrews multiple version of the Rogers–Selberg identity, as well as the Sylvester identity.

We find several new parity results for broken 5-diamond, 7-diamond and 11-diamond partitions.

This paper is concerned with the diophantine problem of finding six integers such that the sum of any two of them is a perfect square. Till now, only one numerical example of such a sextuple has been published. In this paper, we obtain infinitely many examples of sextuples of integers such that the sum of any two of them is a perfect square. These examples include sextuples which have three or four or five distinct integers as well as sextuples in which all the integers are distinct.

In this paper, we will consider the Approximation Lattices for a p-adic number, as defined in a work of de Weger, and construct a generalization called the Cyclotomic Approximation Lattices. In the latter case, we consider approximation by a pair of cyclotomic integers instead of rational ones. This can be useful for studying p-adic continued fractions with cyclotomic integral part. The first section will introduce this work and provide motivations. The second one will give some background theorems on number rings. In the third section, we will recall the work of de Weger with a new proof for Theorem 3.6, the analogue of classical Lagrange's theorem for continued fractions. In the fourth one, we will then see the cyclotomic variant and its analogous properties.

Let K be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let denote an algebraic closure of K. We count points in projective space with given height and generating a quadratic extension of K. If n > 2, we derive an asymptotic estimate for the number of such points as the height tends to infinity. Such estimates are analogous to previous results of Schmidt where the field K is replaced by the field of rational numbers ℚ.

Let s_q(n) denote the sum of the digits of a number n expressed in base q. We study here the ratio for various values of q and α. In 1978, Kenneth B. Stolarsky showed that and that using an explicit construction. We show that for α = 2 and q ≥ 2, the above ratio can in fact be any positive rational number. We also study what happens when α is a rational number that is not an integer, terminating the resulting expression by using the floor function.

Given relatively prime integers a₁,…, a_n, the Frobenius number g(a₁,…, a_n) is defined as the largest integer which cannot be expressed as x₁a₁ +⋯+ x_na_n with x_i nonnegative integers. In this paper, we give the Frobenius number of primitive Pythagorean triples:

Let X be a proper, smooth, and geometrically connected curve over a non-archimedean local field K. In this paper, we relate the component group of the Néron model of the Jacobian of X to the Brauer group of X.

We determine the exceptional sets of hypergeometric functions corresponding to the (2, 4, 6) triangle group by relating them to values of certain quaternionic modular forms at CM points. We prove a result on the number fields generated by exceptional values, and by using modular polynomials we explicitly compute some examples.

In [The trace of Frobenius of elliptic curves and the p-adic gamma function, Pacific J. Math. 261(1) (2013) 219–236], McCarthy defined a function _nG_n[⋯] using the Teichmüller character of finite fields and quotients of the p-adic gamma function. This function extends hypergeometric functions over finite fields to the p-adic setting. In this paper, we give certain transformation formulas for the function _nG_n[⋯] which are not implied from the analogous hypergeometric functions over finite fields.

Let P and Q be nonzero integers. Generalized Fibonacci and Lucas sequences are defined as follows: U₀ = 0, U₁ = 1, and U_n+1 = PU_n + QU_n-1 for n ≥ 1 and V₀ = 2, V₁ = P, and V_n+1 = PV_n + QV_n-1 for n ≥ 1, respectively. For all odd relatively prime values of P and Q such that P ≥ 1, we determine all indices n and m such that V_n = wV_mx² or V_nV_m = wx² with w = 1, 2, 3 or 6 under the assumptions P² + 4Q > 0 and V_m ≠ 1 for all positive integers m.