Vol. LXXXI, 2 (2012)
p. 203 - 209

On integers expressible by some special linear form

A. Dubickas and A. Novikas

Received: November 10, 2011;   Accepted: June 21, 2012

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.

Keywords:  Egyptian fractions; Erdős-Straus conjecture; Sierpiński conjecture; Schinzel's conjecture. 

AMS Subject classification: Primary:  11D68, 11D09, 11Y50  

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Acta Mathematica Universitatis Comenianae
ISSN 0862-9544   (Printed edition)

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Comenius University
842 48 Bratislava, Slovak Republic  

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