Friday, June 28. I found a nice exposition by David Savitt
https://pi.math.cornell.edu/~web401/steve.gauss17gon.pdf
from which this is page 32
David A. Cox, in Galois Theory, gives an account of Gauss's theory of periods
http://zakuski.math.utsa.edu/~jagy/cox_galois_Gaussian_periods.pdf
Cox does only a few examples, but Reuschle filled a book with nothing but.
https://archive.org/details/tafelncomplexer00unkngoog/page/n7/mode/2up
The first few cases where $ z = \frac{x^{23} - y^{23}}{x-y} $ turned out to be prime. I wrote some software to solve $z = u^2 + 23 v^2$ in a reasonable time.
x: 2 y: -1 z: 2796203 z prime ? 2
990^2 + 23 * 281^2 = 2796203
x: 3 y: -1 z: 23535794707 z prime ? 2
118750^2 + 23 * 20253^2 = 23535794707
x: 5 y: 3 z: 5960417405949649 z prime ? 1 ?
38872207^2 + 23 * 13908660^2 = 5960417405949649
x: 6 y: 5 z: 777809294098524691 z prime ? 1 ?
827130254^2 + 23 * 63815235^2 = 777809294098524691
x: 7 y: -1 z: 3421093417510114543 z prime ? 1?
1755643660^2 + 23 * 121370571^2 = 3421093417510114543
x: 7 y: -2 z: 3040971926676589439 z prime ? 1?
1438562808^2 + 23 * 205522555^2 = 3040971926676589439
x: 10 y: 1 z: 11111111111111111111111 z prime ? 1 ?
102063239244^2 + 23 * 5493895055^2 = 11111111111111111111111