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A NOTE ON INTEGERS OF THE FORM 2n+cp

Published online by Cambridge University Press:  05 February 2002

Zhi-Wei Sun  and
Si-Man Yang
Zhi-Wei Sun
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, The People's Republic of China (zwsun@nju.edu.cn)
Si-Man Yang
Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
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Abstract

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In 1950 Erdös proved that if $x\equiv2\,036\,812\ (\mo5\,592\,405)$ and $x\equiv3\ (\mo62)$, then $x$ is not of the form $2^n+p$, where $n$ is a non-negative integer and $p$ is a prime. In this note we present a theorem on integers of the form $2^n+cp$, in particular we completely determine all those integers $c$ relatively prime to $5\,592\,405$ such that the residue class $2\,036\,812(\mo5\,592\,405)$ contains integers of the form $2^n+cp$.

AMS 2000 Mathematics subject classification: Primary 11P32. Secondary 11A07; 11B25; 11B75

Keywords

integers of the form $2^n+cp$cover of $\mathbb{Z}$residue classprimitive prime divisor

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 45 , Issue 1 , February 2002 , pp. 155 - 160
Copyright
Copyright © Edinburgh Mathematical Society 2002