An estimate for the number of consecutive quadratic residues

The problem of estimation of the maximal number H of consecutive integer numbers such that they all are either quadratic residues or quadratic nonresidues modulo a prime number p is considered.     

Catalan and Apéry numbers in residue classes

We estimate character sums with Catalan numbers and middle binomial coefficients modulo a prime p. We use this bound to show that the first at most p^13/2(logp)⁶ elements of each sequence already fall in all residue classes modulo every sufficiently large p, which improves the previously known result requiring p^O(p) elements. We also study, using a different technique, similar questions for sequences satisfying polynomial recurrence relations like the Apéry numbers. We show that such sequences form a finite additive basis modulo p for every sufficiently large prime p.     

Hasse invariants and anomalous primes for elliptic curves with complex multiplication

Let C be an elliptic curve defined over Q. Let p be a prime where C has good reduction. By definition, p is anomalous for C if the Hasse invariant at p is congruent to 1 modulo p. The phenomenon of anomalous primes has been shown by Mazur to be of great interest in the study of rational points in towers of number fields. This paper is devoted to discussing the Hasse invariants and the anomalous primes of elliptic curves admitting complex multiplication. The two special cases Y² = X³ + a₄X and Y² = X³ + a₆ are studied at considerable length. As corollaries, some results in elementary number theory concerning the residue classes of the binomial coefficients (_n²ⁿ) (Resp. (_n³ⁿ)) modulo a prime p = 4n + 1 (resp. p = 6n + 1) are obtained. It is shown that certain classes of elliptic curves admitting complex multiplication do not have any anomalous primes and that others admit only very few anomalous primes.     

On gaps between quadratic non-residues in the Euclidean and Hamming metrics

The authors have recently introduced and studied a modification of the classical number theoretic question about the largest gap between consecutive quadratic non-residues and primitive roots modulo a prime p$p$, where the distances are measured in the Hamming metric on binary representations of integers. Here we continue to study the distribution of such gaps. In particular we prove the upper bound

_?p≤(0.117198…+o(1))logp/log2${?}_p\le (0.117198\dots +o\left(1\right))logp/log2$

for the smallest Hamming weight _?p${?}_p$ among prime quadratic non-residues modulo a sufficiently large prime p$p$. The Burgess bound on the least quadratic non-residue only gives _?p≤(0.15163…+o(1))logp/log2${?}_p\le (0.15163\dots +o\left(1\right))logp/log2$.     

Primitive roots in quadratic fields, II

We consider an analogue of Artin's primitive root conjecture for algebraic numbers which are not units in quadratic fields. Given such an algebraic number α, for a rational prime p which is inert in the field, the maximal possible order of α modulo (p) is p²−1. An extension of Artin's conjecture is that there are infinitely many such inert primes for which this order is maximal. We show that for any choice of 113 algebraic numbers satisfying a certain simple restriction, at least one of the algebraic numbers has order at least for infinitely many inert primes p.     

Congruences modulo 8 for the class numbers of Q(√±p), p ≡ 3(mod 4) a prime

A congruence modulo 8 is proved relating the class numbers of the quadratic fields Q(√p) and Q(√?p), where p is a prime congruent to 3 modulo 4.     

Mixed sums of squares and triangular numbers (III)

In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and only if Tm=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p²=x²+8(y²+z²) for no odd integers x,y,z. We also show that a positive integer cannot be written as a sum of an odd square and two triangular numbers if and only if it is of the form 2Tm(m>0) with 2m+1 having no prime divisor congruent to 3 modulo 4.     

Quadratic residues and non-residues for infinitely many Piatetski-Shapiro primes

In this paper, we prove a quantitative version of the statement that every nonempty finite subset of N+ is a set of quadratic residues for infinitely many primes of the form [nc] with 1 ≤ c ≤243/205. Correspondingly, we can obtain a similar result for the case of quadratic non-residues under reasonable assumptions. These results generalize the previous ones obtained by Wright in certain aspects.     

Small point sets of PG(n, q 3) intersecting each k-subspace in 1 mod q points

The main result of this paper is that point sets of PG(n, q ³), q = p ^h , p ≥ 7 prime, of size less than 3(q ^3(n?k) + 1)/2 intersecting each k-space in 1 modulo q points (these are always small minimal blocking sets with respect to k-spaces) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size less than 3(p ^3(n?k) + 1)/2 with respect to k-spaces are linear. We also give a classification of small linear blocking sets of PG(n, q ³) which meet every (n ? 2)-space in 1 modulo q points.     

Sums of primes and quadratic linear recurrence sequences

Let u be a sequence of positive integers which grows essentially as a geometric progression. We give a criterion on u in terms of its distribution modulo d, d = 1, 2,..., under which the set of positive integers expressible by the sum of a prime number and an element of u has a positive lower density. This criterion is then checked for some second order linear recurrence sequences. It follows, for instance, that the set of positive integers of the form p ＋ [（2 ＋ √3）n], where p is a prime number and n is a positive integer, has a positive lower density. This generalizes a recent result of Enoch Lee. In passing, we show that the periods of linear recurrence sequences of order m modulo a prime number p cannot be ＂too small＂ for most prime numbers p.     

Caliber number of real quadratic fields

We obtain lower bound of caliber number of real quadratic field using splitting primes in K. We find all real quadratic fields of caliber number 1 and find all real quadratic fields of caliber number 2 if d is not 5 modulo 8. In both cases, we don't rely on the assumption on ζK(1/2).     

On the distribution of the primitive roots of a prime

In this paper it is shown that the number of pairs of consecutive primitive roots modulo p is asymptotic to $\text{(p ? 2)(}\text{?(p ? 1)}\text{(p ? 1)}\text{)}{}^2$, and that, for all sufficiently large primes p, there is at least one pair of consecutive primitive roots modulo p. The theorem proved here is a generalization of this proposition. Another one is mentioned in the remarks.     

Density of solutions to quadratic congruences

A classical result in number theory is Dirichlet’s theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly k prime factors for k > 1. Building upon a proof by E.M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree n ≤ x with k prime factors such that a fixed quadratic equation has exactly 2 ^k solutions modulo n.     

A Prime Sensitive Hankel Determinant of Jacobi Symbol Enumerators

We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes if and only if n is composite. If the dimension is a prime p, then the determinant evaluates to a polynomial of degree p − 1 which is the product of a power of p and the generating polynomial of the partial sums of Legendre symbols. The sign of the determinant is determined by the quadratic character of −1 modulo p. The proof of the evaluation makes use of elementary properties of Legendre symbols, quadratic Gauss sums, and orthogonality of trigonometric functions.     

Congruent elliptic curves with non-trivial Shafarevich-Tate groups: Distribution part

Given a large positive number x and a positive integer k, we denote by Qk(x) the set of congruent elliptic curves E(n): y2= z3- n2 z with positive square-free integers n x congruent to one modulo eight,having k prime factors and each prime factor congruent to one modulo four. We obtain the asymptotic formula for the number of congruent elliptic curves E(n)∈ Qk(x) with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)2. We also get a lower bound for the number of E(n)∈ Qk(x)with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)4. The key ingredient of the proof of these results is an independence property of residue symbols. This property roughly says that the number of positive square-free integers n x with k prime factors and residue symbols(quadratic and quartic) among its prime factors being given compatible values does not depend on the actual values.     

Generalized congruence properties of the restricted partition function p(n,m)

Ramanujan-type congruences for the unrestricted partition function p(n) are well known and have been studied in great detail. The existence of Ramanujan-type congruences are virtually unknown for p(n,m), the closely related restricted partition function that enumerates the number of partitions of n into exactly m parts. Let ? be any odd prime. In this paper we establish explicit Ramanujan-type congruences for p(n,?) modulo any power of that prime ? ^α . In addition, we establish general congruence relations for p(n,?) modulo ? ^α for any n.     

On the linear complexity of binary threshold sequences derived from Fermat quotients

We determine the linear complexity of a family of p ²-periodic binary threshold sequences derived from Fermat quotients modulo an odd prime p, where p satisfies ${2^{p-1} \not\equiv 1 ({\rm mod}\, {p^2})}$ . The linear complexity equals p ² ? p or p ² ? 1, depending whether ${p \equiv 1}$ or 3 (mod 4). Our research extends the results from previous work on the linear complexity of the corresponding binary threshold sequences when 2 is a primitive root modulo p ². Moreover, we present a partial result on their linear complexities for primes p with ${2^{p-1} \equiv 1 ({\rm mod} \,{p^2})}$ . However such so called Wieferich primes are very rare.