András Sárközy

Finite addition theorems, I

It is proved that if N, k are positive integers, A ⊂ {1, 2, …, N} and |A |>Nk + 1, then there are integers d, l, m such that 1 ≤ d ≤ k − 1, 1 ≤ l < 118k, and {(m + 1)d, (m + 2)d, …, (m + N)} ⊂ lA.

On Divisibility Properties of Sequences of Integers

Our first joint paper with Erdős appeared in 1966. It was a triple paper with Szemeredi written on divisibility properties of sequences of integers which is one of Erdős’ favorite subjects. Nine further triple papers written on the same subject followed it, and since 1966, we have written altogether 52 joint papers with Erdős. On this special occasion I would like to return to the subject of our very first paper. In Section 2, I will give a survey of the related results, while in Section 3, I will study a further related problem.

A complexity measure for families of binary sequences

In earlier papers finite pseudorandom binary sequences were studied, quantitative measures of pseudorandomness of them were introduced and studied, and large families of “good” pseudorandom sequences were constructed. In certain applications (cryptography) it is not enough to know that a family of “good” pseudorandom binary sequences is large, it is a more important property if it has a “rich”, “complex” structure. Correspondingly, the notion of “f-complexity” of a family of binary sequences is introduced. It is shown that the family of “good” pseudorandom binary sequences constructed earlier is also of high f-complexity. Finally, the cardinality of the smallest family achieving a prescibed f-complexity and multiplicity is estimated.

Large families of pseudorandom sequences of k symbols and their complexity: part II

In earlier papers we introduced the measures of pseudorandomness of finite binary sequences [13], introduced the notion of f–complexity of families of binary sequences, constructed large families of binary sequences with strong PR (= pseudorandom) properties [6], [12], and we showed that one of the earlier constructions can be modified to obtain families with high f–complexity [4]. In another paper [14] we extended the study of pseudorandomness from binary sequences to sequences on k symbols (“letters”). In [14] we also constructed one “good” pseudorandom sequence of a given length on k symbols. However, in the applications we need not only a few good sequences but large families of them, and in certain applications (cryptography) the complexity of the family of these sequences is more important than its size. In this paper our goal is to construct “many” “good” PR sequences on k symbols, to extend the notion of f–complexity to the k symbol case and to study this extended f–complexity concept.

On divisors of binomial coefficients, I

Abstract It is a well-known conjecture that (n2n) is never squarefree if n > 4. It is shown that (n2n) is not squarefree if n > n0.

Unsolved problems in number theory

Abstract68 unsolved problems and conjectures in number theory are presented and brie y discussed. The topics covered are: additive representation functions, the Erdős-Fuchs theorem, multiplicative problems (involving general sequences), additive and multiplicative Sidon sets, hybrid problems (i.e., problems involving both special and general sequences), arithmetic functions, the greatest prime factor func- tion and mixed problems.

On Additive Representation Functions

In this paper we give a short survey of additive representation functions, in particular, on their regularity properties and value distribution. We prove a couple of new results and present many related unsolved problems.

Density and ramsey type results on algebraic equations with restricted solution sets

In earlier papers Sárközy studied the solvability of the equations

On additive properties of general sequences

a + b = cd, a \in \mathcal{A}, b \in \mathcal{B}, c \in \mathcal{C}, d \in \mathcal{D},