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Unambiguous evaluations of bidecic Jacobi and Jacobsthal sums

Published online by Cambridge University Press:  09 April 2009

Ronald J. Evans
Ronald J. Evans
Affiliation:
Department of Mathematics University of California, San Diego La Jolla, California 92093, U.S.A.
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Abstract

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For a class of primes p ≡ 1 (mod 20) for which 2 is a quintic nonresidue, unambiguous evaluations of parameters of bidecic Jacobi and Jacobsthal sums (mod p) are presented, in terms of the partition p = a2+5b2+5c2+5d2, ab = d2–c2–cd. Similar results for sums of others orders have been obtained by E. Lehmer and by K. S. Williams.

Subject classification (Amer. Math. Soc. (MOS) 1970): 10 G 05.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 28 , Issue 2 , September 1979 , pp. 235 - 240
Copyright
Copyright © Australian Mathematical Society 1979

References

Berndt, B. C. and Evans, R. J. (1979a), ‘Sums of Gauss, Jacobi, and Jacobsthal’, J. Number Theory (to appear in 1979).CrossRefGoogle Scholar
Berndt, B. C. and Evans, R. J. (1979b), ‘Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and Brewer’, Illinois J. Math. (to appear in 1979).CrossRefGoogle Scholar
Dickson, L. E. (1935), ‘Cyclotomy, higher congruences and Waring's problem’, Amer. J. Math. 57, 391–424.CrossRefGoogle Scholar
Evans, R. J. (1979), ‘Resolution of sign ambiguities of Jacobi and Jacobsthal sums’, Pacific J. Math. (to appear in 1979).CrossRefGoogle Scholar
Giudici, R. E., Muskat, J. B., and Robinson, S. F. (1972), ‘On the evaluation of Brewer's character sums’, Trans. Amer. Math. Soc. 171, 317–347.Google Scholar
Ireland, K. and Rosen, M. (1972), Elements of number theory (Bogden and Quigley, Tarrytown-on-Hudson).Google Scholar
Lehmer, E. (1951), ‘The quintic character of 2 and 3’, Duke Math. J. 18, 11–18.CrossRefGoogle Scholar
Lehmer, E. (1955), ‘On the number solutions of uk + D ≡ ω (mod p)’, Pacific J. Math. 55, 103–118.Google Scholar
Lehmer, E. (1959), ‘On Euler's criterion’, J. Austral. Math. Soc. 1, 64–70.Google Scholar
Lehmer, E. (1960), ‘On Jacobi functions’, Pacific J. Math. 10, 887–893.Google Scholar
Lehmer, E. (1966), ‘On the quadratic character of the Fibonacci root’, Fibonacci Quart. 4, 135–138.Google Scholar
Lehmer, E. (1972), ‘On some special quartic reciprocity laws’, Acta Arith. 21, 367–377.CrossRefGoogle Scholar
Leonard, P. A. and Williams, K. S. (1976), ‘The eleventh power character of 2’, J. Reine Angew. Math. 286, 213–222.Google Scholar
Muskat, J. B. (1968), ‘On Jacobi sums of certain composite orders’, Trans. Amer. Math. Soc. 134, 483–502.CrossRefGoogle Scholar
Muskat, J. B. and Whiteman, A. L. (1970), ‘The cyclotomic numbers of order twenty’, Acta Arith. 17, 185–216.CrossRefGoogle Scholar
Muskat, J. B. and Zee, Y. C. (1973), ‘Sign ambiguities of Jacobi sums’, Duke Math. J. 40, 13–334.CrossRefGoogle Scholar
Muskat, J. B. and Zee, Y. C. (1975), ‘On the uniqueness of solutions of certain Diophantine equations’, Proc. Amer. Math. Soc. 49, 13–19.CrossRefGoogle Scholar
Nashier, B. S. and Rajwade, A. R. (1977), ‘Determination of a unique solution of the quadratic partition for primes p ≡ 1 (mod 7)’, Pacific J. Math. 72, 513–521.CrossRefGoogle Scholar
Rajwade, A. R. (1975), ‘Notes on the congruence y2 ≡ x5–a (mod p)’, Enseign. Math. 21, 49–56.Google Scholar
Williams, K. S. (1975a), ‘On Euler's criterion for cubic nonresidues’, Proc. Amer. Math. Soc. 49, 277–283.Google Scholar
Williams, K. S. (1975b), ‘On Euler's criterion for quintic nonresidues’, Pacific J. Math. 61, 543–550.CrossRefGoogle Scholar
Williams, K. S. (1979) (unpublished).Google Scholar