Avoiding triples in arithmetic progression ∗

On Triples in Arithmetic Progression

On rainbow 4-term arithmetic progressions

{sl Let $[n]={1,dots, n}$ be colored in $k$ colors. A rainbow AP$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. Conlon, Jungi'{c} and Radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow AP(4) free, when $n$ is even. Based on their construction, we show that such a coloring of $[4n]$...

The Green-Tao Theorem on Primes in Arithmetic Progression: A Dynamical Point of View

A long standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao...

Markoff-Rosenberger triples in arithmetic progression

Article history: Received 22 March 2012 Accepted 12 November 2012 Available online 27 November 2012

Diophantine m-tuples with elements in arithmetic progressions

In this paper, we consider the problem of existence of Diophantine m-tuples which are (not necessarily consecutive) elements of an arithmetic progression. We show that for n ≥ 3 there does not exist a Diophantine quintuple {a, b, c, d, e} such that a ≡ b ≡ c ≡ d ≡ e (mod n). On the other hand, for any positive integer n there exist infinitely many Diophantine triples {a, b, c} such that a ≡ b ≡...