On The Partition Regularity of $ax+by=c{w}^m{z}^n$

Abstract

Csikvári, Gyarmati, and Sárközy showed that the equation $x+y=z^2$ is not partition regular (PR) over $\mathbb{N}$ and asked if the equation $x+y=wz$ is PR over $\mathbb{N}$. Bergelson and Hindman independently answered this question in the positive. We generalize this result by giving a partial classification of the $a,b,c\in \mathbb{Z}\setminus \{0\}$ and $m,n\in \mathbb{N}$ for which the equation $ax+by=c{w}^m{z}^n$ is PR over $\mathbb{Z}\setminus \{0\}$. We show that if $m,n\ge 2$, then $ax+by=c{w}^m{z}^n$ is PR over $\mathbb{Z}\setminus \{0\}$ if and only if $a+b=0$. Next, we show that if $n$ is odd, then the equation $ax+by=cw{z}^n$ is PR over $\mathbb{Z}\setminus \{0\}$ if and only if one of $\frac{a}{c},\frac{b}{c},$ or $\frac{a+b}{c}$ is an $n$th power in $\mathbb{Q}$. We come close to a similar characterization of the partition regularity of $ax+by=cw{z}^n$ over $\mathbb{Z}\setminus \{0\}$ for even $n$, and we examine some equations whose partition regularity remain unknown, such as $16x+17y=w{z}^8$. In order to show that the equation $ax+by=cw{z}^n$ is not PR over $\mathbb{Z}\setminus \{0\}$ for certain values of $a,b,c,$ and $n$, we prove a partial generalization of the criteria of Grunwald and Wang for when $\alpha \in \mathbb{Z}$ is an $n$th power modulo every prime $p$. In particular, we show that for any odd $n$ and any $\alpha ,\beta ,\gamma \in \mathbb{Q}$ that are not $n$th powers, there exist infinitely many primes $p\in \mathbb{N}$ for which none of $\alpha ,\beta ,$ and $\gamma$ are $n$th powers modulo $p$. Similarly, we show that for any even $n$ and any $\alpha ,\beta ,\gamma \in \mathbb{Q}$ that are not $\frac{n}{2}$th powers, with one not an $\frac{n}{4}$th power if $4|n$, there exist infinitely many primes $p\in \mathbb{N}$ for which $\alpha ,\beta ,$ and $\gamma$ are not $n$th powers modulo $p$. Part of the abstract was removed here.

arXiv comment52 pages originally, 54 pages on arxiv, this version has incorporated the referee's comments and has corrected many typos

References in corpus (1)

• On Rado conditions for nonlinear Diophantine equations