The Jacobi sums of order twenty-two
HTML articles powered by AMS MathViewer
- by Yun-cheng Zee PDF
- Proc. Amer. Math. Soc. 28 (1971), 25-31 Request permission
Abstract:
This paper completes the analysis of the Jacobi sums of order 22 outlined by L. E. Dickson. Furthermore, it is shown that a prime $p$ of the form $22f + 1$ has the binary quadratic decomposition $4p = {u^2} + 11{v^2}$ and that certain Jacobi sums can be evaluated in terms of $u$ and $v$, where $u$ satisfies $u \equiv 9\pmod 11$.References
- L. D. Baumert and H. Fredricksen, The cyclotomic numbers of order eighteen with applications to difference sets, Math. Comp. 21 (1967), 204–219. MR 223322, DOI 10.1090/S0025-5718-1967-0223322-5 H. Davenport and H. Hasse, Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen, J. Reine Angew. Math. 172 (1934), 151-182.
- L. E. Dickson, Cyclotomy, Higher Congruences, and Waring’s Problem, Amer. J. Math. 57 (1935), no. 2, 391–424. MR 1507083, DOI 10.2307/2371217
- L. E. Dickson, Cyclotomy and trinomial congruences, Trans. Amer. Math. Soc. 37 (1935), no. 3, 363–380. MR 1501791, DOI 10.1090/S0002-9947-1935-1501791-3
- L. E. Dickson, Cyclotomy when $e$ is composite, Trans. Amer. Math. Soc. 38 (1935), no. 2, 187–200. MR 1501808, DOI 10.1090/S0002-9947-1935-1501808-6
- T. Estermann, On the sign of the Gaussian sum, J. London Math. Soc. 20 (1945), 66–67. MR 15412, DOI 10.1112/jlms/s1-20.2.66
- Edmund Landau, Elementary number theory, Chelsea Publishing Co., New York, N.Y., 1958. Translated by J. E. Goodman. MR 0092794
- J. B. Muskat, The cyclotomic numbers of order fourteen, Acta Arith. 11 (1965/66), 263–279. MR 193081, DOI 10.4064/aa-11-3-263-279
- Joseph B. Muskat, On Jacobi sums of certain composite orders, Trans. Amer. Math. Soc. 134 (1968), 483–502. MR 232752, DOI 10.1090/S0002-9947-1968-0232752-4
- Joseph B. Muskat and Albert L. Whiteman, The cyclotomic numbers of order twenty, Acta Arith. 17 (1970), 185–216. MR 268151, DOI 10.4064/aa-17-2-185-216
- Trygve Nagell, Introduction to number theory, 2nd ed., Chelsea Publishing Co., New York, 1964. MR 0174513 H. J. S. Smith, Report on the theory of numbers, Chelsea, New York, 1965.
- Albert Leon Whiteman, The cyclotomic numbers of order sixteen, Trans. Amer. Math. Soc. 86 (1957), 401–413. MR 92807, DOI 10.1090/S0002-9947-1957-0092807-7
- Albert Leon Whiteman, The cyclotomic numbers of order ten, Proc. Sympos. Appl. Math., Vol. 10, American Mathematical Society, Providence, R.I., 1960, pp. 95–111. MR 0113851
- A. L. Whiteman, The cyclotomic numbers of order twelve, Acta Arith. 6 (1960), 53–76. MR 118709, DOI 10.4064/aa-6-1-53-76
- Yun-cheng Zee, The Jacobi sums of orders thirteen and sixty and related quadratic decompositions, Math. Z. 115 (1970), 259–272. MR 262202, DOI 10.1007/BF01129976
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 28 (1971), 25-31
- MSC: Primary 10.66
- DOI: https://doi.org/10.1090/S0002-9939-1971-0272748-5
- MathSciNet review: 0272748