The cyclotomic numbers of order sixteen
Authors:
Ronald J. Evans and Jay Roderick Hill
Journal:
Math. Comp. 33 (1979), 827835
MSC:
Primary 10G15; Secondary 10A40, 12C20
MathSciNet review:
521298
Fulltext PDF Free Access
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.
 [1]
Pierre
Barrucand and Harvey
Cohn, Note on primes of type 𝑥²+32𝑦²,
class number, and residuacity, J. Reine Angew. Math.
238 (1969), 67–70. MR 0249396
(40 #2641)
 [2]
L.
D. Baumert and H.
Fredricksen, The cyclotomic numbers of order
eighteen with applications to difference sets, Math. Comp. 21 (1967), 204–219. MR 0223322
(36 #6370), http://dx.doi.org/10.1090/S00255718196702233225
 [3]
Bruce
C. Berndt and Ronald
J. Evans, Sums of Gauss, Jacobi, and Jacobsthal, J. Number
Theory 11 (1979), no. 3, S. Chowla Anniversary Issue,
349–398. MR
544263 (81j:10054), http://dx.doi.org/10.1016/0022314X(79)900088
 [4]
Bruce
C. Berndt and Ronald
J. Evans, Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and
Brewer, Illinois J. Math. 23 (1979), no. 3,
374–437. MR
537798 (81j:10055)
 [5]
L.
E. Dickson, Cyclotomy, Higher Congruences, and Waring’s
Problem, Amer. J. Math. 57 (1935), no. 2,
391–424. MR
1507083, http://dx.doi.org/10.2307/2371217
 [6]
Ronald
J. Evans, Bioctic Gauss sums and sixteenth power residue difference
sets, Acta Arith. 38 (1980/81), no. 1,
37–46. MR
574123 (81h:10004)
 [7]
Ronald
J. Evans, Resolution of sign ambiguities in Jacobi and Jacobsthal
sums, Pacific J. Math. 81 (1979), no. 1,
71–80. MR
543734 (81a:10051)
 [8]
Reinaldo
E. Giudici, Joseph
B. Muskat, and Stanley
F. Robinson, On the evaluation of Brewer’s
character sums, Trans. Amer. Math. Soc. 171 (1972), 317–347.
MR
0306122 (46 #5249), http://dx.doi.org/10.1090/S00029947197203061225
 [9]
Helmut
Hasse, Der 2ⁿte Potenzcharakter von 2 im Körper der
2ⁿten Einheitswurzeln, Rend. Circ. Mat. Palermo (2)
7 (1958), 185–244 (German). MR 0105401
(21 #4143)
 [10]
Emma
Lehmer, On cyclotomic numbers of order sixteen, Canadian J.
Math. 6 (1954), 449–454. MR 0063398
(16,115b)
 [11]
Emma
Lehmer, On the number of solutions of
𝑢^{𝑘}+𝐷≡𝑤²(\𝑚𝑜𝑑𝑝),
Pacific J. Math. 5 (1955), 103–118. MR 0067918
(16,798b)
 [12]
Emma
Lehmer, On Jacobi functions, Pacific J. Math.
10 (1960), 887–893. MR 0113852
(22 #4683)
 [13]
Philip
A. Leonard and Kenneth
S. Williams, The cyclotomic numbers of order eleven, Acta
Arith. 26 (1974/75), no. 4, 365–383. MR 0376513
(51 #12688)
 [14]
Philip
A. Leonard and Kenneth
S. Williams, The cyclotomic numbers of order
seven, Proc. Amer. Math. Soc. 51 (1975), 295–300. MR 0371868
(51 #8085), http://dx.doi.org/10.1090/S00029939197503718688
 [15]
J.
B. Muskat, The cyclotomic numbers of order fourteen, Acta
Arith. 11 (1965/1966), 263–279. MR 0193081
(33 #1302)
 [16]
Joseph
B. Muskat and Albert
L. Whiteman, The cyclotomic numbers of order twenty, Acta
Arith. 17 (1970), 185–216. MR 0268151
(42 #3050)
 [17]
Albert
Leon Whiteman, The cyclotomic numbers of order
sixteen, Trans. Amer. Math. Soc. 86 (1957), 401–413. MR 0092807
(19,1160h), http://dx.doi.org/10.1090/S00029947195700928077
 [18]
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
(22 #4682)
 [19]
A.
L. Whiteman, The cyclotomic numbers of order twelve, Acta
Arith. 6 (1960), 53–76. MR 0118709
(22 #9480)
 [1]
 P. BARRUCAND & H. COHN, "Note on primes of the form , class number, and residuacity," J. Reine Angew. Math., v. 238, 1969, pp. 6770. 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. 204219. 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. 391424. 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. 317347. MR 0306122 (46:5249)
 [9]
 H. HASSE, "Der te Potenzcharakter von 2 im Körper der ten Einheitswurzeln," Rend. Circ. Mat. Palermo Ser. II, v. 7, 1958, pp. 185243. MR 0105401 (21:4143)
 [10]
 E. LEHMER, "On the cyclotomic numbers of order sixteen," Canad. J. Math., v. 6, 1954, pp. 449454. MR 0063398 (16:115b)
 [11]
 E. LEHMER, "On the number of solutions of ," Pacific J. Math., v. 5, 1955, pp. 103118. MR 0067918 (16:798b)
 [12]
 E. LEHMER, "On Jacobi functions," Pacific J. Math., v. 10, 1960, pp. 887893. MR 0113852 (22:4683)
 [13]
 P. A. LEONARD & K. S. WILLIAMS, "The cyclotomic numbers of order eleven," Acta Arith., v. 26, 1975, pp. 367383. 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. 295300. MR 0371868 (51:8085)
 [15]
 J. B. MUSKAT, "The cyclotomic numbers of order fourteen," Acta Arith., v. 11, 1966, pp. 263279. MR 0193081 (33:1302)
 [16]
 J. B. MUSKAT & A. L. WHITEMAN, "The cyclotomic numbers of order twenty," Acta Arith., v. 17, 1970, pp. 185216. MR 0268151 (42:3050)
 [17]
 A. L. WHITEMAN, "The cyclotomic numbers of order sixteen," Trans. Amer. Math. Soc., v. 86, 1957, pp. 401413. 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. 95111. MR 0113851 (22:4682)
 [19]
 A. L. WHITEMAN, "The cyclotomic numbers of order twelve," Acta Arith., v. 6, 1960, pp. 5376. 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:
http://dx.doi.org/10.1090/S00255718197905212982
PII:
S 00255718(1979)05212982
Keywords:
Cyclotomic numbers of order sixteen,
Jacobi sums,
Jacobsthal sums,
sign ambiguities
Article copyright:
© Copyright 1979 American Mathematical Society
