Prime divisors of sparse integers

We obtain a lower bound on the number of prime divisors of integers whose g-ary expansion contains a fixed number of nonzero digits. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the multidimensional distribution of the subset sum generator of pseudorandom numbers

We show that for a random choice of the parameters, the subset sum pseudorandom number generator produces a sequence of uniformly and independently distributed pseudorandom numbers. The result can be useful for both cryptographic and quasi-Monte Carlo applications and relies on bounds of exponential sums.

     

On the Distribution of Power Residues and Primitive Elements in Some Nonlinear Recurring Sequences

It is shown that the method of estimation of exponential sumswith nonlinear recurring sequences, invented by the authorsin a recent series of works, can be applied to estimating sumsof multiplicative characters as well. As a consequence, resultsare obtained about the distribution of power residues and primitiveelements in such sequences. 2000 Mathematics Subject Classification11B37, 11L40 (primary), 11A07, 11A15, 11T24 (secondary).     

On congruences with products of variables from short intervals and applications

We obtain upper bounds on the number of solutions to congruences of the type (x ₁ + s)... (x _ν + s) ≡ (y ₁ + s)... (x _ν + s) ? 0 (mod p) modulo a prime p with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M.Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. Friedlander and H. Iwaniec and some results of M.-C. Chang and A.A. Karatsuba on character sums twisted with the divisor function.     

Squarefree parts of discriminants of trinomials

We obtain a lower bound on the number of distinct squarefree parts of the discriminants n ⁿ + (?1) ⁿ (n?1) ^n-1 of trinomials \({X^n - X - 1\in \mathbb{Z}[X]}\) for \({1 \leqslant n \leqslant N}\) .     

Multiplicative congruences with variables from short intervals

Recently, several bounds have been obtained on the number of solutions of congruences of the type $$({x_1} + s) \cdots ({x_v} + s) \equiv ({y_1} + s) \cdots ({y_v} + s)\not \equiv 0{\text{ (mod }}p{\text{),}}$$ where p is prime and variables take values in some short interval. Here, for almost all p and all s and also for a fixed p and almost all s, we derive stronger bounds. We also use similar ideas to show that for almost all p, one can always find an element of a large order in any rather short interval.     

Average distribution of prime ideals in families of number fields

We view an algebraic curve over ℚ as providing a one-parameter family of number fields and obtain bounds for the average value of some standard prime ideal counting functions over these families which are better than averaging the standard estimates for these functions.      

Bounds of trilinear sums with Kloosterman fractions

Archiv der Mathematik - We obtain a new bound for trilinear exponential sums with Kloosterman fractions which in some ranges of parameters improves that of S. Bettin and V. Chandee (2018). We also...     

Estimates for Wieferich numbers

We define Wieferich numbers to be those odd integers w≥3 that satisfy the congruence 2 ^φ(w)≡1 (mod  w ²). It is clear that the distribution of Wieferich numbers is closely related to the distribution of Wieferich primes, and we give some quantitative forms of this statement. We establish several unconditional asymptotic results about Wieferich numbers; analogous results for the set of Wieferich primes remain out of reach. Finally, we consider several modifications of the above definition and demonstrate that our methods apply to such sets of integers as well. During the preparation of this paper, W.B. was supported in part by NSF grant DMS-0070628, F.L. was supported in part by grants SEP-CONACYT 37259-E and 37260-E, and I.S. was supported in part by ARC grant DP0211459.     

On the values of the divisor function

For a positive integer n we let τ(n) denote the number of its positive divisors. In this paper, we obtain lower and upper bounds for the average value of the ratio τ(n + 1)/τ(n) as n ranges through positive integers in the interval [1,x]. We also study the cardinality of the sets {τ(p − 1) : p ≤ x prime} and {τ(2ⁿ − 1) : n ≤ x}. Authors’ addresses: Florian Luca, Instituto de Matemáticas, Universidad Nacional Autónoma'de'México, C.P. 58089, Morelia, Michoacán, México; Igor E. Shparlinski, Department of Computing, Macquarie University, Sydney, NSW 2109, Australia