In this paper we generalize and extend work by Languasco, Perreli and Zaccagnini on the Montgomery–Hooley theorem in short intervals. We are thus able to obtain asymptotic formulae for the mean-square error in the distribution of general sequences in arithmetic progressions in short intervals. As applications we consider lower and upper bound approximations to the characteristic function of the primes in a short interval and an average over short intervals for the von Mangoldt function.

We investigate the first and second moments of shifted convolutions of the generalized divisor function d3(n).

We study the distribution of the size of Selmer groups and Tate–Shafarevich groups arising from a 2-isogeny and its dual 2-isogeny for elliptic curves En:y2=x3−n3. We show that the 2-ranks of these groups all follow the same distribution. The result also implies that the mean value of the 2-rank of the corresponding Tate–Shafarevich groups for square-free positive integers n≤X is as X→∞. This is quite different from quadratic twists of elliptic curves with full 2-torsion points over ℚ [M. Xiong and A. Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math. 219 (2008), 523–553], where one Tate–Shafarevich group is almost always trivial while the other is much larger.

Brizolis asked for which primes p greater than 3 there exists a pair (g,h) such that h is a fixed point of the discrete exponential map with base g, or equivalently h is a fixed point of the discrete logarithm with base g. Various authors have contributed to the understanding of this problem. In this paper, we use p-adic methods, primarily Hensel’s lemma and p-adic interpolation, to count fixed points, two-cycles, collisions, and solutions to related equations modulo powers of a prime p.

Let r be an integer greater than 1, and let A be a finite, nonempty set of nonzero integers. We obtain a lower bound for the number of positive squarefree integers n, up to x, for which the products ∏ p∣n(p+a) (over primes p) are perfect rth powers for all the integers a in A. Also, in the cases where A={−1} and A={+1}, we will obtain a lower bound for the number of such n with exactly r distinct prime factors.

The existence of products of three pairwise coprime integers is investigated in short intervals of the form . A general theorem is proved which shows that such integer products exist provided there is a bound on the product of any two of them. A particular case of relevance to elliptic curve cryptography, where all three integers are of order , is presented as a corollary to this result.

In this paper we solve the equation f(g(x))=f(x)hm(x) where f(x), g(x) and h(x) are unknown polynomials with coefficients in an arbitrary field K, f(x) is nonconstant and separable, deg g≥2, the polynomial g(x) has nonzero derivative g′(x)≠0in K[x]and the integer m≥2is not divisible by the characteristic of the field K. We prove that this equation has no solutions if deg f≥3 . If deg f=2 , we prove that m=2and give all solutions explicitly in terms of Chebyshev polynomials. The Diophantine applications for such polynomials f(x) , g(x) , h(x)with coefficients in ℚ or ℤ are considered in the context of the conjecture of Cassaigne et al. on the values of Liouville’s λ function at points f(r) , r∈ℚ.

We give upper and lower bounds on the count of positive integers n ≤ x dividing the nth term of a non-degenerate linearly recurrent sequence with simple roots.

Let a be an integer different from 0, ±1, or a perfect square. We consider a conjecture of Erdős which states that #{p:ℓa(p)=r}≪εrε for any ε>0, where ℓa(p) is the order of a modulo p. In particular, we see what this conjecture says about Artin’s primitive root conjecture and compare it to the generalized Riemann hypothesis and the ABC conjecture. We also extend work of Goldfeld related to divisors of p+a and the order of a modulo p.

Let be a commutative algebraic group defined over a number field K. For a prime ℘ in K where has good reduction, let N℘,n be the number of n-torsion points of the reduction of modulo ℘ where n is a positive integer. When is of dimension one and n is relatively prime to a fixed finite set of primes depending on , we determine the average values of N℘,n as the prime ℘ varies. This average value as a function of n always agrees with a divisor function.

In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author.

Extending classical results of Nair and Tenenbaum, we provide general, sharp upper bounds for sums of the type where x,y,u,v have comparable logarithms, F belongs to a class defined by a weak form of sub-multiplicativity, and the Qj are arbitrary binary forms. A specific feature of the results is that the bounds are uniform within the F-class and that, as in a recent version given by Henriot, the dependency with respect to the coefficients of the Qj is made explicit. These estimates play a crucial rôle in the proof, published separately by the authors, of Manin’s conjecture for Châtelet surfaces.

Let k,r≥2 be two integers. We prove an asymptotic formula for the number of k-free values of the r variables polynomial t1⋯tr−1 over [1,x]r∩ℤr.

Given an integer d≥2, a d-normal number, or simply a normal number, is an irrational number whosed-ary expansion is such that any preassigned sequence, of length k≥1, taken within this expansion occurs at the expected limiting frequency, namely 1/dk. Answering questions raised by Igor Shparlinski, we show that 0.P(2)P(3)P(4)…P(n)… and 0.P(2+1)P(3+1)P(5+1)…P(p+1)…, where P(n) stands for the largest prime factor of n, are both normal numbers.

Let k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetic function gk,a,b for any positive integer n by

If we let a = 1 and b = 0, then gk,a,b becomes the arithmetic function that was previously introduced by Farhi. Farhi proved that gk,1,0 is periodic and that k! is a period. Hong and Yang improved Farhi's period k! to lcm(1, 2, … , k) and conjectured that (lcm(1, 2, … , k, k + 1))/(k + 1) divides the smallest period of gk,1,0. Recently, Farhi and Kane proved this conjecture and determined the smallest period of gk,1,0. For the general integers a ≥ 1 and b ≥ 0, it is natural to ask the following interesting question: is gk,a,b periodic? If so, what is the smallest period of gk,a,b? We first show that the arithmetic function gk,a,b is periodic. Subsequently, we provide detailed p-adic analysis of the periodic function gk,a,b. Finally, we determine the smallest period of gk,a,b. Our result extends the Farhi–Kane Theorem from the set of positive integers to general arithmetic progressions.

Given a prime p, the Fermat quotient qp(u) of u with gcd (u,p)=1 is defined by the conditions We derive a new bound on multiplicative character sums with Fermat quotients qp(ℓ) at prime arguments ℓ.

A Diophantine m-tuple is a set A of m positive integers such that ab+1 is a perfect square for every pair a,b of distinct elements of A. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by x. In doing so, we extend two existing tools in ways that might be of independent interest. The Erdős–Turán inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also adapt an argument of Hooley on the equidistribution of solutions of polynomial congruences to handle reducible quadratic polynomials.

For any irreducible quadratic polynomial f(x) in ℤ[x], we obtain the estimate log l.c.m.(f(1),…,f(n))=nlog n+Bn+o(n), where B is a constant depending on f.

A Lehmer number is a composite positive integer n such that ϕ(n)|n − 1. In this paper, we show that given a positive integer g > 1 there are at most finitely many Lehmer numbers which are repunits in base g and they are all effectively computable. Our method is effective and we illustrate it by showing that there is no such Lehmer number when g ∈ [2, 1000].

We prove two results about the asymptotic formula for The first result is for x7/12+ε≤h≤x and h/(log x)B≤Q≤h, where ε,B>0 are arbitrary constants. For the second result we assume that the Generalized Riemann Hypothesis holds and we obtain a stronger error term and a better uniformity on h.