A strange pigeon-hole principle

The first proposition and its corollary are a transfiguration of Dirichlet's pigeon-hole principle. They are applied to show that a wide variety of sequences display arbitrarily large patterns of sums, differences, higher differences, etc. Among these, we include sequences of primes in arithmetic progressions, of powerful integers, sequences of integers with radical index having a prescribed lower bound, and many others. We also deal with patterns in iterated sequences of primes, patterns of gaps between primes, patterns of values of Euler's φ-function, or their gaps, as well as patterns related to the sequence of Carmichael numbers.

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On the strength of several versions of Dirichlet's (“pigeon-hole”-)principle in the sense of first-order logic

Archive for Mathematical Logic ◽

10.1007/bf02011634 ◽

1981 ◽

Vol 21 (1) ◽

pp. 69-76 ◽

Cited By ~ 1

Author(s):

T. Twer

Keyword(s):

Order Logic ◽

First Order Logic ◽

Pigeon Hole Principle ◽

First Order

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Applications of the Pigeon-Hole Principle

The Mathematical Gazette ◽

10.2307/3618300 ◽

1995 ◽

Vol 79 (485) ◽

pp. 286 ◽

Cited By ~ 1

Author(s):

Kiril Bankov

Keyword(s):

Pigeon Hole Principle

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Count(q) versus the Pigeon-Hole Principle

BRICS Report Series ◽

10.7146/brics.v1i26.21640 ◽

1994 ◽

Vol 1 (26) ◽

Author(s):

Søren Riis

Keyword(s):

Complete Classification ◽

Pigeon Hole Principle ◽

Prime Factors ◽

Open Question

For each p <= 2 there exist a model M* of IDelta_{0}(alpha) which satisfies the Count(p) principle. Furthermore if p contain all prime factors of q there exist n, r in M* and a bijective map f in Set(M*) mapping {1, 2, ..., n} onto {1,2,...,n+q^r}. <br /> <br />A corollary is a complete classification of the Count(q) versus Count(p) problem. Another corollary solves an open question by M. Ajtai.

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