A positive quadratic form is $(k,\ell)$-universal if it represents all the numbers $kx+\ell$ where $x$ a non-negative integer, and almost but finitely many of them. We prove that for any $k,\ell$ such $k\nmid\ell$ there exists an diagonal ternary form. also conjecture are only primes $p$ which $(p,\ell)$-universal (for $\ell<p$) we show results computer experiments speak in favor conjecture.

A 3–coloring of the elements of an abelian group is said to be rainbow–free if there is no 3–term arithmetic progression with its members having pairwise distinct colors. We give a structural characterization of rainbow–free colorings of abelian groups. This characterization proves a conjecture of Jungić et al. on the size of the smallest chromatic class of a rainbow–free 3–coloring of cyclic g...

We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions. A complete characterization is given for the infinite family of triangles with sides in arithmetic progression. We show that there are no triangles with sides in geometric progression. We also show that apart from the square there are no cyclic quadrilaterals whose si...

The usual product $m\cdot n$ on $\mathbb{Z}$ can be viewed as the sum of $n$ terms an arithmetic progression whose first term is $a_{1}=m-n+1$ and difference $d=2$. Generalizing this idea, we define new similar mappings, consider arithmetics that enable us to extend Furstenberg's theorem infinitude primes. We also review classic conjectures in arithmetics. Finally, make important extensions mai...

Let A be a subset of the primes. Let δP (N) = |{n ∈ A : n ≤ N}| |{n prime : n ≤ N}| . We prove that, if δP (N) ≥ C log log logN (log logN)1/3 for N ≥ N0, where C and N0 are absolute constants, then A ∩ [1, N ] contains a non-trivial three-term arithmetic progression. This improves on Green’s result [Gr], which needs δP (N) ≥ C s log log log log logN log log log logN .

For a prime p, we call a non-empty subset S of the group Fp balanced if every element of S is the midterm of a three-term arithmetic progression, contained in S. A result of Browkin, Divǐs and Schinzel implies that the size of a balanced subset of Fp is at least log2 p+1. In this paper we present an efficient algorithm which yields a balanced set of size (1 + o(1)) log2 p as p grows.

We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the socalled Hardy-Littlewood majorant property. We derive this from a rather more general result which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a r...