ON KUREPA ’ S PROBLEMS IN NUMBER THEORY Dedicated to the memory of Prof . - Duro

We discuss some problems in number theory posed by-Duro Kurepa, including the so-called left factorial hypothesis that an odd prime p does not divide 0! + 1! + · · · + (p − 1)!. 1. Introduction-D. Kurepa posed several problems in number theory that drew attention of many workers in number theory. Certainly, the most known of his problems is the so called left factorial hypothesis, which is still an open problem. However, Kurepa asked several other questions that are less known, but we think that they are interesting as well. The aim of this paper is to review some of these problems, and to present some of the known results concerning them. We shall assume the following notation. We shall denote by N the set of natural numbers (nonnegative integers), N + denotes positive integers, while Z n denotes the ring of integers modulo n. The greatest common divisor of integers a and b is denoted by (a, b). The Galois field of p elements, where p is a prime, is denoted by GF(p). If m and n are integers, by rest(m, n) we shall denote the remainder obtained from division of m by n.