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


[![enter image description here][1]][1]



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  


  [1]: https://i.sstatic.net/zflMmk5n.png