A theorem of Dekking in the combinatorics of words implies that there exists an injective order-preserving transformation f : {0, 1, . . . , n} → {0, 1, . . . , 2n} with the restriction f(i + 1) ≤ f(i) + 2 such that for every 5-term arithmetic progression P its image f(P ) is not an arithmetic progression. In this paper we consider symmetric sets in place of arithmetic progressions and prove lo...

Let σ(k) denote the maximum of the number of squares in a+b, . . . , a+kb as we vary over positive integers a and b. Erdős conjectured that σ(k) = o(k) which Szemerédi [30] elegantly proved as follows: If there are more than δk squares amongst the integers a+b, . . . , a+kb (where k is sufficiently large) then there exists four indices 1 ≤ i1 < i2 < i3 < i4 ≤ k in arithmetic progression such th...

Abstract We consider the reduction of an elliptic curve defined over rational numbers modulo primes in a given arithmetic progression and investigate how often subgroup points this reduced is cyclic.

Dirichlet’s theorem on primes in arithmetic sequences states that in any arithmetic progression m,m + k, m + 2k, m + 3k, . . ., there are infinitely many primes, provided that (m, k) = 1. Euler first conjectured a result of this form, claiming that every arithmetic progression beginning with 1 contained an infinitude of primes. The theorem as stated was conjectured by Gauss, and proved by Diric...