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

Csikv'ari, Gyarmati, and S'ark"ozy showed that the equation $x+y=z^2$ is not partition regular (PR) over $\mathbb{N}$ and asked if the equation $x+y = wz$isPRover$$\mathbb{N}$$. 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}$$forwhichtheequation$ax+by = cw^mz^n$isPRover$$\mathbb{Z}$\setminus{0}$\mathrm{.}Weshowthatif$m,n \ge 2$,then$ax+by = cw^mz^n$isPRover$$\mathbb{Z}$\setminus{0}$ifandonlyif$a+b = 0$. Next, we show that if $n$ is odd, then the equation $ax+by=cw{z}^n$ is PR over $\mathbb{Z}\setminus \{0\}$ if and only if one of $\frac{a}{c},\frac{b}{c},$ or $\frac{a+b}{c}$ is an $n$th power in $\mathbb{Q}$. We come close to a similar characterization of the partition regularity of $ax+by=cw{z}^n$ over $\mathbb{Z}\setminus \{0\}$ for even $n$, and we examine some equations whose partition regularity remain unknown, such as $16x+17y=w{z}^8$. In order to show that the equation $ax+by=cw{z}^n$ is not PR over $\mathbb{Z}\setminus \{0\}$ for certain values of $a,b,c,$ and $n$, we prove a partial generalization of the criteria of Grunwald and Wang for when $\alpha \in $\mathbb{Z}$$isan$n$thpowermoduloeveryprime$p$. 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}$$thatarenot$\frac{n}{2}$thpowers,withonenotan$\frac{n}{4}$thpowerif$4|n$,thereexistinfinitelymanyprimes$p \in $\mathbb{N}$$forwhich$\alpha,\beta,$and$\gamma$arenot$n$thpowersmodulo$p$. Part of the abstract was removed here.