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2017-10-03, 02:00   #8
carpetpool

"Sam"
Nov 2016

26·5 Posts

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 n-th 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 qf, 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?