Determining the least prime in an arithmetic progression is already known to be a very hard problem. The classic result is due to Kulik. In a sense it is satisfying (given the difficulty of the problem), but it is far from what is suspected.
What might be more interesting here, is, for a given n, noting the class number h = h(n) for the field of nth roots of unity, and the number k of primes congruent to 1 (mod n) you have to go through until you find one that is the norm of principal ideal. My guess is, k will not be terribly larger than h, but it could be much smaller.
BTW, if n is a prime power q^{f}, then 1  \zeta_{n} is a principal generator of the (unique) prime ideal of norm q.
Last fiddled with by Dr Sardonicus on 20171002 at 14:17
