Abstract
In Sárközy (2001) the third author of this paper presented two conjectures on the additive decomposability of the sequence of ”smooth” (or ”friable”) numbers. Elsholtz and Harper (2015) proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of S-unit equations).
Highlights
A, B, C, . . . denote sets of non-negative integers, and their counting functions are denoted by A(X), B(X), C(X), . . . so that e.g
The set of the positive integers is denoted by N, and we write N∪{0} = N0 for the set of non-negative integers
A finite or infinite set A of non-negative integers is said to be a-reducible if it has an additive decomposition
Summary
Denote (usually infinite) sets of non-negative integers, and their counting functions are denoted by A(X), B(X), C(X), . The set of the positive integers is denoted by N, and we write N∪{0} = N0 for the set of non-negative integers. The set of rational numbers is denoted by Q
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