The cyclotomic numbers of order sixteen
Albert Leon Whiteman · Transactions of the American Mathematical Society · 1957
at least for some (i, j). To answer this question she undertook the following experiment on the SWAC (National Bureau of Standards Western Automatic Computer). The cyclotomic constants of order sixteen were computed for eight primes p of the form 32n?1 for which 2 is not a biquadratic residue. She found that (1.2) is not satisfied for any (i, j)l6 when the signs of y and b are taken in accordance with the results on cyclotomic constants of order eight. A similar computation for primes p of the form 32n + 17 also led to a negative result. The SWAC experiment leaves open the question of determining if the equation (1.2) can be satisfied for any prime p for which 2 is a biquadratic residue. In the present paper this is answered in the affirmative for six of the cyclotomic constants. The following formulas involving only p, a and x are derived. Let p = 16f+1 be a prime. If the integer m is defined by the congruence gm= 2 (mod p), then