Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

The cyclotomic numbers of order sixteen


Authors: Ronald J. Evans and Jay Roderick Hill
Journal: Math. Comp. 33 (1979), 827-835
MSC: Primary 10G15; Secondary 10A40, 12C20
DOI: https://doi.org/10.1090/S0025-5718-1979-0521298-2
MathSciNet review: 521298
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A complete table of 408 formulas for cyclotomic numbers of order sixteen is presented. Each number is expressed as a linear combination of parameters of quartic, octic, and bioctic Jacobi sums. Recent applications of these formulas are discussed.


References [Enhancements On Off] (What's this?)

  • [1] P. BARRUCAND & H. COHN, "Note on primes of the form $ {x^2} + 32{y^2}$, class number, and residuacity," J. Reine Angew. Math., v. 238, 1969, pp. 67-70. MR 0249396 (40:2641)
  • [2] L. BAUMERT & H. FREDERICKSEN, "The cyclotomic numbers of order eighteen with applications to difference sets," Math. Comp., v. 21, 1967, pp. 204-219. MR 0223322 (36:6370)
  • [3] B. C. BERNDT & R. J. EVANS, "Sums of Gauss, Jacobi, and Jacobsthal," J. Number Theory. (To appear.) MR 544263 (81j:10054)
  • [4] B. C. BERNDT & R. J. EVANS, "Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and Brewer," Illinois J. Math. (To appear.) MR 537798 (81j:10055)
  • [5] L. E. DICKSON, "Cyclotomy, higher congruences, and Waring's problem," Amer. J. Math., v. 57, 1935, pp. 391-424. MR 1507083
  • [6] R. J. EVANS, "Bioctic Gauss sums and sixteenth power residue difference sets," Acta Arith. (To appear.) MR 574123 (81h:10004)
  • [7] R. J. EVANS, "Resolution of sign ambiguities of Jacobi and Jacobsthal sums," Pacific J. Math. (To appear.) MR 543734 (81a:10051)
  • [8] R. GIUDICI, J. MUSKAT & S. ROBINSON, "On the evaluation of Brewer's character sums," Trans. Amer. Math. Soc., v. 171, 1972, pp. 317-347. MR 0306122 (46:5249)
  • [9] H. HASSE, "Der $ {2^n}$-te Potenzcharakter von 2 im Körper der $ {2^n}$-ten Einheitswurzeln," Rend. Circ. Mat. Palermo Ser. II, v. 7, 1958, pp. 185-243. MR 0105401 (21:4143)
  • [10] E. LEHMER, "On the cyclotomic numbers of order sixteen," Canad. J. Math., v. 6, 1954, pp. 449-454. MR 0063398 (16:115b)
  • [11] E. LEHMER, "On the number of solutions of $ {u^k} + D \equiv {w^2}\;\pmod p$," Pacific J. Math., v. 5, 1955, pp. 103-118. MR 0067918 (16:798b)
  • [12] E. LEHMER, "On Jacobi functions," Pacific J. Math., v. 10, 1960, pp. 887-893. MR 0113852 (22:4683)
  • [13] P. A. LEONARD & K. S. WILLIAMS, "The cyclotomic numbers of order eleven," Acta Arith., v. 26, 1975, pp. 367-383. MR 0376513 (51:12688)
  • [14] P. A. LEONARD & K. S. WILLIAMS, "The cyclotomic numbers of order seven," Proc. Amer. Math. Soc., v. 51, 1975, pp. 295-300. MR 0371868 (51:8085)
  • [15] J. B. MUSKAT, "The cyclotomic numbers of order fourteen," Acta Arith., v. 11, 1966, pp. 263-279. MR 0193081 (33:1302)
  • [16] J. B. MUSKAT & A. L. WHITEMAN, "The cyclotomic numbers of order twenty," Acta Arith., v. 17, 1970, pp. 185-216. MR 0268151 (42:3050)
  • [17] A. L. WHITEMAN, "The cyclotomic numbers of order sixteen," Trans. Amer. Math. Soc., v. 86, 1957, pp. 401-413. MR 0092807 (19:1160h)
  • [18] A. L. WHITEMAN, "The cyclotomic numbers of order ten," Proc. Sympos. Appl. Math., vol. 10, Amer. Math. Soc., Providence, R. I., 1960, pp. 95-111. MR 0113851 (22:4682)
  • [19] A. L. WHITEMAN, "The cyclotomic numbers of order twelve," Acta Arith., v. 6, 1960, pp. 53-76. MR 0118709 (22:9480)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10G15, 10A40, 12C20

Retrieve articles in all journals with MSC: 10G15, 10A40, 12C20


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1979-0521298-2
Keywords: Cyclotomic numbers of order sixteen, Jacobi sums, Jacobsthal sums, sign ambiguities
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society