Cyclotomy of order twice a prime (Q1071811)

The paper aims to relate the ordinary Gaussian cyclotomy for \(p=2ef+1\) (e,p primes) to the ''generalized cyclotomy'' (considered recently by Evans) of order e in which the periods are linear combinations of the Gaussian periods. Use is made of a certain theorem proved by the author on Kloosterman sums. Results on the so called ''exceptional primes'' and ''special primes'' are included. In the latter half of the paper results of the first half are applied to the case \(2e=10\) making use of the Dickson diophantine system \[ 16p=x^ 2+50 u^ 2+50 v^ 2+125 w^ 2,\quad etc. \] to obtain some interesting results on exceptional primes and special primes.