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

Bruce M. Landman, Kevin Ventullo

Research output: Contribution to journalArticle

1 Citation (Scopus)

### 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 language English (US) 207-214 8 Utilitas Mathematica 82 Published - Jul 1 2010 Yes

Coloring
Accessibility
Integer
Prime factor
Colouring
Exceed
Odd
Term

### ASJC Scopus subject areas

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

### Cite this

In: Utilitas Mathematica, Vol. 82, 01.07.2010, p. 207-214.

Research output: Contribution to journalArticle

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