Quote:
Originally Posted by Dr Sardonicus
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.

Yes Dr. Sardonicus, and this would make for a very interesting problem.
Given the cyclotomic field Kn = Q(\zeta_{n}), the smallest prime p > n which is a norm of principal ideal are:
3, 7, 5, 11, 7, 29, 17, 19, 11, 23, 13, 53, 29, 31, 17, 103, 19, 191, 41, 43, 23, 599, 73, 101, 53, 109, 29, 4931, 31, 5953, 97, 67, 103, 71, 73, 32783, 191, 157, 41, 101107, 43, 178021, 89, 181, 599,...
The one for K47 and larger n in Kn is unknown.
I am unable to find the next terms to this sequence. Do you care to help with it further?