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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 . 10^{9}.
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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