Random conjecture For any numbers $a$ and $b$ so that $a>b$ and $\text{gcd}(a, b)=1$, there exists a $c\in\mathbb{N^+}$ so that $a+bc$ is prime.
I've only just tested this out with a few numbers, but I'm curious as to what you guys have to say.
$$100+21\cdot3
\\97+96\cdot1
\\97+50\cdot2
\\482914+15435\cdot5
\\500009+500000\cdot4$$

There is more than one $c$. There are infinitely many. See Dirichlet's theorem.
The proof requires some background in Analytic Number Theory, although.
What Dirichlet proved is that certain sum indexed by the primes $a+kb$ where $\gcd(a,b)=1$ and $k$ goes from $1$ to $\infty$ diverges. And if a sum divereges it must have infinitely many terms.
