Fix a prime $p$.

I want to get $k<p$ primes $p_1<\dots<p_k$ such that at every $i\in\{1,\dots,k\}$ we have $$p_i\equiv (2i+1+c)\bmod p$$ where $c$ is fixed and $2k+1+c<p$ holds.

For a given $p$ what is the maximum $k$ I can get?

For a given $p$ and given $k$ what is the minimum $p_1$ and $p_k$ that can achieve this?

How small can $\prod_{i=1}^kp_i$ be?

What are the smallest ratios $\frac{p_1}{p}$ and $\frac{p_k}{p}$?

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