On a problem of Erdös related to common factor differences

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Abstract

Let n be a positive integer. The factor-difference set D(n) of n is the set of absolute values d of the differences between the factors of any factorization of n as a product of two integers. Erdos and Rosenfeld [The factor-difference set of integers, Acta Arith. 79(4) (1997) 353-359] ask whether for every positive integer n there exist integers N1 < ⋯ < Nn such that |∩i=1nD(N i)|≥ n, and prove this is true when n = 2. Urroz [A note on a conjecture of Erdos and Rosenfeld, J. Number Theory 78(1) (1999) 140-143] shows the result true for n = 3. The ideas of this paper can be extended, and here, we show the result true for n = 4 by proving there are infinitely many sets of four integers with four common factor differences.

Original languageEnglish (US)
JournalInternational Journal of Number Theory
DOIs
StateAccepted/In press - Jan 1 2019

Keywords

  • common factor difference
  • elliptic curve
  • Erdös problem

ASJC Scopus subject areas

  • Algebra and Number Theory

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