Integers have many interesting properties. In this paper it will be shown that, for an arbitrary nonconstant arithmetic progression {an}TM=l of positive integers (denoted by N), either {an}TM=l contains infinitely many palindromic numbers or else 10|aw for every n GN. (This result is a generalization of the theorem concerning the existence of palindromic multiples, cf. [2].) More generally, for...

How few three-term arithmetic progressions can a subset S ⊆ ZN := Z/NZ have if |S| ≥ υN? (that is, S has density at least υ). Varnavides [4] showed that this number of arithmetic-progressions is at least c(υ)N for sufficiently large integers N ; and, it is well-known that determining good lower bounds for c(υ) > 0 is at the same level of depth as Erdös’s famous conjecture about whether a subset...

Addressing a question of Cameron and Erdős, we show that, for infinitely many values of n, the number of subsets of {1, 2, . . . , n} that do not contain a k-term arithmetic progression is at most 2O(rk(n)), where rk(n) is the maximum cardinality of a subset of {1, 2, . . . , n} without a k-term arithmetic progression. This bound is optimal up to a constant factor in the exponent. For all value...

Buchmann and Pethő [5] observed that following algebraic integer 10 + 9α + 8α + 7α + 6α + 5α + 4α, with α = 3 is a unit. Since the coefficients form an arithmetic progressions they have found a solution to the Diophantine equation (1) NK/Q(x0 + αx1 + · · ·+ x6α) = ±1, such that (x0, . . . , x6) ∈ Z is an arithmetic progression. Recently Bérczes and Pethő [3] considered the Diophantine equation ...

It is shown that for each integer k > 3, there exists a set Sk of positive integers containing no arithmetic progression of k terms, such that 2„6Si \/n > (1 e)k log A:, with a finite number of exceptional k for each real e > 0. This result is shown to be superior to that attainable with other sets in the literature, in particular Rankin's sets &(k), which have the highest known asymptotic dens...

Let U (n) denote the maximal length arithmetic progression in a non-uniform random subset of {0, 1} n , where 1 appears with probability pn. By using dependency graph and Stein-Chen method, we show that U (n) − cn ln n converges in law to an extreme type distribution with ln pn = −2/cn. Similar result holds for W (n) , the maximal length aperiodic arithmetic progression (mod n). An arithmetic p...