Andrzej Schinzel

On reducible trinomials, III

It is shown that if a trinomial has a trinomial factor then under certain conditions the cofactor is irreducible.

On the factors of Stern polynomials

It is proved that the Stern polynomial with a prime index does not have a properndivisor over Q of degree less than 4.

On some open problems of number theory

I shall present here 14 open problems of number theory arranged chronologically from Antiquity to 1952, giving each time the best known partial result now and in 1952. Antiquity has left us a definition of a perfect number and two problems concerning perfect numbers. Definition 1. A positive integer n is perfect if it is the sum of its all proper divisors. Problem 1. Do there exist infinitely many even perfect numbers? Euclid’s Elements contains a formula for even perfect numbers, which in the modern notation looks 2p−1(2p − 1), where 2 − 1 is a prime. Euler proved that all even perfect numbers are given by this formula. Till now we know 49 even perfect numbers, the greatest being 274207280(274207281 − 1), in January 1952 we knew only 12 even perfect numbers, the greatest being 2126(2127−1) (see [5], and [2, B1]). Problem 2. Does there exist an odd perfect number? L. E. Dickson proved in 1913 that for every given k there exist only finitely many odd perfect numbers with exactly k distinct prime factors. P. Nielsen proved in 2003 that such numbers are less than 24k (see [7, p. 15]). We now know that for odd perfect numbers n, n > 10300, k = ω(n) ≥ 9, in 1952 we knew that n > 5 · 105, ω(n) ≥ 6 (see ibid.). Mathematics Subject Classification: 11-02. A lecture given at the University of Debrecen on March 9, 2017, slightly enlarged and modified. 256 Andrzej Schinzel In order to reach the next open problem, we have to mention amicable numbers, studied first in the Middle Ages. Definition 2. Positive integers m < n are amicable if

Polynomials dertermined by a few of their coefficients

Abstract We prove some results which indicate that a monic polynomial over a field of characteristic zero with exactly κ distinct zeros may be determined up to finitely many possibilities by any κ of its nonzero proper coefficients.