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...

In this note we observe that in the hyper-graph removal lemma the edge removal can be done in a way that the symmetries of the original hyper-graph remain preserved. As an application we prove the following generalization of Szemerédi’s Theorem on arithmetic progressions. If in an Abelian group A there are sets S1, S2 . . . , St such that the number of arithmetic progressions x1, x2, . . . , xt...

The Green-Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves that there exist arbitrarily long strings of consecutive primes that lie in any arithmetic progression that contains infinitely many primes. Using the technique...

Let [Formula: see text] be the von Mangoldt function, let an integer and counting function for Goldbach numbers with summands in arithmetic progression modulo a common text]. We prove asymptotic formula weighted average, Cesàro weight of order text], this function. Our result is uniform suitable range

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...

Problem 1. Suppose you have seven dice—each a different color of the rainbow; otherwise the dice are standard, with faces numbered 1 to 6. A roll is a sequence specifying a value for each die in rainbow (ROYGBIV) order. For example, one roll is .3; 1; 6; 1; 4; 5; 2/ indicating that the red die showed a 3, the orange die showed 1, the yellow 6,. . . . For the problems below, describe a bijection...

In this note we are interested in the problem of whether or not every increasing sequence of positive integers x1x2x3 · · · with bounded gaps must contain a double 3-term arithmetic progression, i.e., three terms xi, xj , and xk such that i+ k = 2j and xi+xk = 2xj . We consider a few variations of the problem, discuss some related properties of double arithmetic progressions, and present severa...

A path in an edge-colored graph G, where adjacent edges may be colored the same, is called a rainbow path if no two edges of G are colored the same. For a κ-connected graph G and an integer k with 1 ≤ k ≤ κ, the rainbow k-connectivity rck(G) of G is defined as the minimum integer j for which there exists a j-edgecoloring of G such that every two distinct vertices of G are connected by k interna...

We prove several results regarding edge-colored complete graphs and rainbow cycles, cycles with no color appearing on more than one edge. We settle a question posed by Ball, Pultr, and Vojtěchovský [BPV05] by showing that if such a coloring does not contain a rainbow cycle of length n, where n is odd, then it also does not contain a rainbow cycle of length m for all m greater than 2n. In additi...