While stumbling upon this question, I came across this interesting Theorem of Daniel Shiu:
Theorem (Shiu, 2000): Let $p_n$ denote the $n$-th prime. $\forall k \in \mathbb{N}$ and $a, q \in \mathbb{N}, q \ge 3$ having $\gcd(a,q) = 1$ there exist ‘strings’ of congruent primes $p_{n+1}, p_{n+2}, \dots p_{n+k}$ such that $p_{n+1} \equiv p_{n+2} \equiv \dots \equiv p_{n+k} \equiv a \pmod{q}.$
Question: Given $a,q, k$ with $\gcd(a, q) = 1$, is there a way to obtain $p_{n+1}, p_{n+2}, \dots p_{n+k}$ efficiently?
I brute-forced a search for $(k, q) = (4, 7)$ and found:
$$ \begin{align} 2982827 & \equiv & 2982841 & \equiv & 2982869 & \equiv & 2982883 & \equiv 1 \pmod{7} \\ 894301 & \equiv & 894329 & \equiv & 894343 & \equiv & 894371 & \equiv 2 \pmod{7} \\ 1734827 & \equiv & 1734841 & \equiv & 1734869 & \equiv & 1734883 & \equiv 3 \pmod{7} \\ 1651787 & \equiv & 1651801 & \equiv & 1651829 & \equiv & 1651843 & \equiv 4 \pmod{7} \\ 1819067 & \equiv & 1819109 & \equiv & 1819123 & \equiv & 1819151 & \equiv 5 \pmod{7} \\ 4148087 & \equiv & 4148101 & \equiv & 4148129 & \equiv & 4148143 & \equiv 6 \pmod{7} \end{align} $$
Is trial division guaranteed to get the sequence of primes quickly for longer strings of primes or are there any clever tricks to use?
Unfortunately, I do not have access to the full paper and unable to check if the paper itself provides a constructive method to find the prime sequence.
References:
D. K. L. Shiu, Strings of Congruent Primes, Journal of the London Mathematical Society, Volume 61, Issue 2, April 2000, Pages 359–373, https://doi.org/10.1112/S0024610799007863
