Let $S=k*m+p$ be any number where $k,m\in\mathbb{N}$ and $p$ is a prime number.
My claim is the following:
There exists some nondecreasing function $f:\mathbb{N}\mapsto\mathbb{N}$ with $f(m)\to\infty$ and $\frac{f(m)}{m}\to 0$ for $m\to\infty$, such that for any $m\in\mathbb{N}$ the number of primes that satisfy
- $p\leq f(m)$ and
- for at least one value of $k\in\mathbb{N}$, $S$ is a square number that is coprime to $m$
is less than some constant $\Phi\in\mathbb{N}$.
I posted this about an hour ago but it was written poorly with many mistakes and I hope that this is a much better way of presentation.
My guess is that $\Phi$ could be huge, if the claim holds true. Although I think that there is probably a counter example that I am not seeing. If there is not, then I would suspect $f(m)=log(m)$ defined for $m\geq T$ for some $T\in\mathbb{N}$ to be a candidate for this kind of slowly growing function.