Conjecture:
Given $m,n\in\mathbb N^+$, one odd and one even, there are two
primes $p,q$ such that $|mp-nq|=\gcd(m,n)$.
I hope MSE can determine its validity.
From time to time, when testing my growing math packages BigZ and Forthmath, I recognize some patterns which I can't prove or disprove (or even have the ambition to). I post them here with the hope that it will not annoy too much. I hope you can bear with it.
Best Answer
Put $d=\gcd(m,n)$. By Bézout's identity there are two numbers $a,b\in\mathbb N$ with $\gcd(a,b)=1$ such that $|ma-nb|=d$. Then by linear Diophantine equations solving, all solutions to $|mx-ny|=d$ are of the form $x=a+kn'$, $y=b+km'$, where $n'=n/d, m'=m/d$. Therefore you are searching for the two arithmetic progressions $x_k=a+kn', y_k=b+km'$ to yield simultaneous primes for the same value $k$.
Since you asked for kewyords: This is conjectured (but not proven) to be true, and is called Dickson's conjecture, which is a particular case of the famous Schinzel's hypothesis H, which generalizes to polynomials of arbitrary degree (seeing arithmetic progressions as polynomials of degree $1$). Schinzel's hypothesis H encompasses many other important difficult conjectures, as the twin primes conjecture. The only result proven in this regard is Dirichlet's theorem, for just one arithmetic progression. A related conjecture is Bunyakowsky's conjecture, for only one polynomial of arbitrary degree.