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Sequence and perfect squares

Source: Problem 5, Polish NO 1990

September 30, 2005
limitnumber theory unsolvednumber theory

Problem Statement

Suppose that (an)(a_n) is a sequence of positive integers such that lim⁡n→∞nan=0\lim\limits_{n\to \infty} \dfrac{n}{a_n}=0 Prove that there exists kk such that there are at least 19901990 perfect squares between a1+a2+...+aka_1 + a_2 + ... + a_k and a1+a2+...+ak+1a_1 + a_2 + ... + a_{k+1}.
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