### Abstract

For a set D of positive integers, a sequence {a_{1} < a _{2} < ⋯ a_{k}} is called an k-term D-diffsequence if a_{i} - a_{i-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) |
---|---|

Pages (from-to) | 207-214 |

Number of pages | 8 |

Journal | Utilitas Mathematica |

Volume | 82 |

State | Published - Jul 1 2010 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Applied Mathematics
- Statistics and Probability

### Cite this

*Utilitas Mathematica*,

*82*, 207-214.

**Avoiding monochromatic sequences with gaps in a fixed translation of the primes.** / Landman, Bruce M.; Ventullo, Kevin.

Research output: Contribution to journal › Article

*Utilitas Mathematica*, vol. 82, pp. 207-214.

}

TY - JOUR

T1 - Avoiding monochromatic sequences with gaps in a fixed translation of the primes

AU - Landman, Bruce M.

AU - Ventullo, Kevin

PY - 2010/7/1

Y1 - 2010/7/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=79251640192&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79251640192&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:79251640192

VL - 82

SP - 207

EP - 214

JO - Utilitas Mathematica

JF - Utilitas Mathematica

SN - 0315-3681

ER -