Abstract In this paper we determine the number and typical structure of sets integers with bounded doubling. particular, improving recent results Green Morris, Mazur, show that following holds for every fixed $\lambda> 2$ $k \geqslant (\log n)^4$: if $\omega \rightarrow \infty $ as $n (arbitrarily slowly), then almost all $A \subset [n]$ $|A| = k$ $|A + A| \leqslant \lambda are contained...

and we consider two equations related by such a change of variables to represent the same curve (equivalently, we will deal with elliptic curves up to so-called Weierstrass changes of variables). Consider P0, . . . , Pn ∈ E(K), with Pi = (xi, yi) such that x0, . . . , xn is an arithmetic progression. We say that P0, . . . , Pn are in x-arithmetic progression (x-a.p.) and also say that E has an ...

The partitions of a positive integer n in which the parts are in arithmetic progression possess interesting combinatorial properties that distinguish them from other classes of partitions. We exhibit the properties by analyzing partitions with respect to a fixed length of the arithmetic progressions. We also address an open question concerning the number of integers k for which there is a k-par...

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a,q)=1$. Denote by $\mathcal{P}_2(a,q)$ the least $\mathcal{P}_2$ which satisfies $\mathcal{P}_2\equiv a\pmod q$. In this paper, it is proved for sufficiently large $q$, there holds \begin{equation*} \mathcal{P}_2(a,q)\ll q^{...

In this paper, we investigate arithmetic progressions in the polygonal numbers with a fixed number of sides. We first show that four-term arithmetic progressions cannot exist. We then describe explicitly how to find all three-term arithmetic progressions. Finally, we show that not only are there infinitely many three-term arithmetic progressions, but that there are infinitely many three-term ar...