# The Graham conjecture implies the Erdos-Turan conjecture

## TL;DR

This paper demonstrates that proving the Graham conjecture for a specific grid size implies the truth of the Erdős-Turán conjecture for certain progression lengths, linking two significant conjectures in number theory.

## Contribution

It establishes a conditional implication from the Graham conjecture to the Erdős-Turán conjecture, connecting progressions and grid structures in number theory.

## Key findings

- Proves that Graham conjecture for s implies Erdős-Turán for 2s-1
- Links grid patterns in
extbf{N}× extbf{N} to arithmetic progressions in extbf{N}
- Provides a new approach to relate two longstanding conjectures

## Abstract

Erd\"{o}s and Tur\'{a}n once conjectured that any set $A\subset\mathbb{N}$ with $\sum_{a\in A}{1}/{a}=\infty$ should contain infinitely many progressions of arbitrary length $k\geq3$. For the two-dimensional case Graham conjectured that if $B\subset \mathbb{N}\times\mathbb{N}$ satisfies $$\sum\limits_{(x,y)\in B}\frac{1}{x^2+y^2}=\infty,$$ then for any $s\geq2$, $B$ contains an $s\times s$ axes-parallel grid. In this paper it is shown that if the Graham conjecture is true for some $s\geq2$, then the Erd\"{o}s-Tur\'{a}n conjecture is true for $k=2s-1$.