A problem of Erdös concerning partitions

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.

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 |...

Problem of Erdös – Vershik for Golden Ratio

On a phenomenon of Turán concerning the summands of partitions

Turán [12] proved that for almost all pairs of partitions of an integer, the proportion of common parts is very high, that is greater than 1 2 − ε with ε > 0 arbitrarily small. In this paper we prove that this surprising phenomenon persists when we look only at the summands in a fixed arithmetic progression.

On a Problem of P . Erdös and S. Stein

is called a covering system if every integer satisfies at least one of the congruences (1) . An old conjecture of P . Erdös states that for every integer a there is a covering system with n l = c. Selfridge and others settled this question for c < 8 . The general case is still unsettled and seems difficult . A system (1) is called disjoint if every integer satisfies at most one of the congruenc...