Sumsets being squares

Alon, Angel, Benjamini and Lubetzky 1] recently studied an old problem of Euler on sumsets for which all elements of A+B are integer squares. Improving their result we prove: 1. There exists a set A of 3 positive integers and a corresponding set B?0,N] with |B|?(logN)^15/17, such that all elements of A+B are perfect squares. 2. There exists a set A of 3 integers and a corresponding set B?0,N] with |B|?(logN)^9/11, such that all elements of the sets A, B and A+B are perfect squares. The proofs make use of suitably constructed elliptic curves of high rank.