Avoiding monochromatic sequences with gaps in a fixed translation of the primes

Bruce M. Landman, Kevin Ventullo

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

For a set D of positive integers, a sequence {a1 < a 2 < ⋯ ak} is called an k-term D-diffsequence if ai - ai-1 ∈ D for all i ∈ {2,..., k}. For a positive integer r, a set of positive integers D is r-accessible if every r-coloring of ℤ+ has arbitrarily long monochromatic D-diffsequences. The largest r such that D is r-accessible is called the degree of accessibility of D. It is already known that each odd translation of the set of primes, P + t, is 2-accessible. We offer new results on the accessibility of translations the primes. The main result is that for any c ≥ 2, the degree of accessibility of P + c does not exceed the smallest prime factor of c.

Original languageEnglish (US)
Pages (from-to)207-214
Number of pages8
JournalUtilitas Mathematica
Volume82
StatePublished - Jul 1 2010
Externally publishedYes

ASJC Scopus subject areas

  • Statistics, Probability and Uncertainty
  • Applied Mathematics
  • Statistics and Probability

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