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On Cyclotomic Numbers of Order Sixteen

Published online by Cambridge University Press:  20 November 2018

Emma Lehmer
Emma Lehmer*
Affiliation:
Berkeley, California
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It has been shown by Dickson (1) that if (i, j)8 is the number of solutions of

(mod p),

then 64(i,j)8 is expressible for each i,j, as a linear combination with integer coefficients of p, x, y, a, and b where

,

and

a ≡ b ≡ 1 (mod 4),

while the sign of y and b depends on the choice of the primitive root g. There are actually four sets of such formulas depending on whether p is of the form 16n + 1 or 16n + 9 and whether 2 is a quartic residue or not.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 6 , 1954 , pp. 449 - 454
Copyright
Copyright © Canadian Mathematical Society 1954

References

1. Dickson, L. E., Cyclotomy, higher congruences and Waring's problem, Amer. J. Math. 57 (1935), 391–242.Google Scholar
2. Lehmer, Emma, On the number of solutions of uk + D ≡ w2 (mod p), to appear shortly in the Pacific Journal of Mathematics.Google Scholar