On special primes (Q1061759)

The author finds special primes for the two quartics arising from the cyclotomy of Kloosterman sums (for \(e=8)\) studied by \textit{D. H. Lehmer} and the author [Acta Arith. 12, 385-407 (1967; Zbl 0149.286); ibid. 14, 89-111 (1968; Zbl 0192.394)]; a special prime q being a prime which divides the discriminant of a general period polynomial of degree e associated with the prime \(p=ef+1\) but q neither an eth power mod p nor a divisor of any value of this polynomial. The author also shows that there are no such special primes for \(e<8\). These primes are in any case, very rare.