Erdös problem and quadratic equation

The Inverse Erdös-Heilbronn Problem

The famous Erdős-Heilbronn conjecture (first proved by Dias da Silva and Hamidoune in 1994) asserts that if A is a subset of Z/pZ, the cyclic group of the integers modulo a prime p, then |A +̂ A| > min{2 |A| − 3, p}. The bound is sharp, as is shown by choosing A to be an arithmetic progression. A natural inverse result was proven by Karolyi in 2005: if A ⊂ Z/pZ contains at least 5 elements and |...

On a problem of Erdös and Graham

In this paper we provide bounds for the size of the solutions of the Diophantine equation x(x+ 1)(x+ 2)(x+ 3)(x+ k)(x+ k+ 1)(x+ k+ 2)(x+ k+ 3) = y, where 4 ≤ k ∈ N is a parameter. We also determine all integral solutions for 1 ≤ k ≤ 10.

Problem of Erdös – Vershik for Golden Ratio

Erdös graphs resolve Fine's canonicity problem

We show that there exist 2א0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order defi...

Intuitionistic fuzzy stability of a quadratic and quartic functional equation

In this paper, we prove the generalized Hyers--Ulamstability of a quadratic and quartic functional equation inintuitionistic fuzzy Banach spaces.