On integers expressible by some special linear form

Abstract

Let E(4) be the set of positive integers expressible by the form 4M - d, where M is a multiple of the product ab and d is a divisor of the sum a + b of two positive integers a, b. We show that the set E(4) does not contain perfect squares and three exceptional positive integers 288, 336, 4545 and verify that E(4) contains all other positive integers up to 2 . 109. We conjecture that there are no other exceptional integers. This would imply the Erdős-Straus conjecture asserting that each number of the form 4/n, where n ³ 2 is a positive integer, is the sum of three unit fractions 1/x + 1/y + 1/z. We also discuss similar problems for sets E(t), where t ³ 3, consisting of positive integers expressible by the form tM - d. The set E(5) is related to a conjecture of Sierpiński, whereas the set E(t), where t is any integer greater than or equal to 4, is related to the most general in this context conjecture of Schinzel.