Journal of the Ramanujan Mathematical Society

Goutam Pal ¹

Affiliations
1 Dept. of Mathematics, RCC-Institute of Information Technology, South Canal Road, Beliaghata, Kolkata-700015, IN

Abstract

Let S_p be the set of positive integers containing no p terms in arithmetic progression generated by the greedy algorithm. We show that for p an odd prime, the set S_p does not maximize the sum ∑_a∈A 1/a , where the set A runs over all sequences of positive integers containing no p terms in arithmetic progression.

Full Text

     

Goutam Pal ¹

Affiliations
1 Department of Mathematics, RCC-Institute of Information Technology, South Canal Road, Beliaghata, Kolkata 700 015, IN

Abstract

A famous conjecture of Erd¨os asserts that if A is a subset of the positive integers having the property that ∑a∈A 1/a = ∞, then A must contain arithmetic progressions of arbitrarily large length. We use topological ideas to show that if this conjecture is true then there must exist a subset M_p of positive integers with no p terms in arithmetic progression with the property that among all such subsets, M_p maximizes the sum of the reciprocals of its elements.

Full Text