Abstract
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.
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.
| Original language | English |
|---|---|
| Pages (from-to) | 353-357 |
| Journal | Acta Mathematica Hungarica |
| Volume | 141 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - 2013 |
Fields of Expertise
- Information, Communication & Computing
Treatment code (Nähere Zuordnung)
- Basic - Fundamental (Grundlagenforschung)