Explicit criteria for quintic residuacity
Author:
Kenneth S. Williams
Journal:
Math. Comp. 30 (1976), 847853
MSC:
Primary 10A15
MathSciNet review:
0412089
Fulltext PDF Free Access
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Abstract: Let p be a . Necessary and sufficient conditions are determined for the prime to be a quintic residue of p. The results for are known, the rest are new.
 [1]
C. E. BICKMORE, "On the numerical factors of . II," Messenger Math., v. 26, 1897, pp. 138.
 [2]
A. J. C. CUNNINGHAM & T. GOSSET, "4tic and 3tic residuacity tables," Messenger Math., v. 50, 1920, pp. 130.
 [3]
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
 [4]
P.
D. T. A. Elliott, A problem of Erdős concerning power
residue sums, Acta Arith. 13 (1967/1968),
131–149. MR 0220689
(36 #3741)
 [5]
K. G. J. JACOBI, "De residuis cubicis commentatio numerosa," J. Reine Angew. Math., v. 2, 1827, pp. 6669.
 [6]
E. E. KUMMER, "Über die Divisoren gewisser Formen der Zahlen, welche aus der Theorie der Kreistheilung enstehen," J. Reine Angew. Math., v. 30, 1846, pp. 107116.
 [7]
Emma
Lehmer, The quintic character of 2 and 3, Duke Math. J.
18 (1951), 11–18. MR 0040338
(12,677a)
 [8]
Emma
Lehmer, Criteria for cubic and quartic residuacity,
Mathematika 5 (1958), 20–29. MR 0095162
(20 #1668)
 [9]
Emma
Lehmer, Artiads characterized, J. Math. Anal. Appl.
15 (1966), 118–131. MR 0201377
(34 #1261)
 [10]
Emma
Lehmer, On the divisors of the discriminant of the period
equation, Amer. J. Math. 90 (1968), 375–379. MR 0227133
(37 #2718)
 [11]
Horst
von Lienen, Primzahlen als achte Potenzreste, J. Reine Angew.
Math. 266 (1974), 107–117 (German). MR 0340160
(49 #4916)
 [12]
G.
B. Mathews, Theory of numbers, 2nd ed, Chelsea Publishing Co.,
New York, 1961. MR 0126402
(23 #A3698)
 [13]
Joseph
B. Muskat, Criteria for solvability of certain congruences,
Canad. J. Math. 16 (1964), 343–352. MR 0163871
(29 #1170)
 [14]
J.
B. Muskat, On the solvability of
𝑥^{𝑒}≡𝑒(𝑚𝑜𝑑𝑝),
Pacific J. Math. 14 (1964), 257–260. MR 0159781
(28 #2997)
 [1]
 C. E. BICKMORE, "On the numerical factors of . II," Messenger Math., v. 26, 1897, pp. 138.
 [2]
 A. J. C. CUNNINGHAM & T. GOSSET, "4tic and 3tic residuacity tables," Messenger Math., v. 50, 1920, pp. 130.
 [3]
 L. E. DICKSON, "Cyclotomy, higher congruences, and Waring's problem," Amer. J. Math., v. 57, 1935, pp. 391424. MR 1507083
 [4]
 P. D. T. A. ELLIOTT, "A problem of Erdös concerning power residue sums," Acta Arith., v. 13, 1967/68, pp. 131149; Corrigendum, ibid., v. 14, 1967/68, p. 437. MR 36 #3741; 37 #4031. MR 0220689 (36:3741)
 [5]
 K. G. J. JACOBI, "De residuis cubicis commentatio numerosa," J. Reine Angew. Math., v. 2, 1827, pp. 6669.
 [6]
 E. E. KUMMER, "Über die Divisoren gewisser Formen der Zahlen, welche aus der Theorie der Kreistheilung enstehen," J. Reine Angew. Math., v. 30, 1846, pp. 107116.
 [7]
 EMMA LEHMER, "The quintic character of 2 and 3," Duke Math. J., v. 18, 1951, pp. 1118. MR 12, 677. MR 0040338 (12:677a)
 [8]
 EMMA LEHMER, "Criteria for cubic and quartic residuacity," Mathematika, v. 5, 1958, pp. 2029. MR 20 #1668. MR 0095162 (20:1668)
 [9]
 EMMA LEHMER, "Artiads characterized," J. Math. Anal. Appl., v. 15, 1966, pp. 118131. MR 34 #1261. MR 0201377 (34:1261)
 [10]
 EMMA LEHMER, "On the divisors of the discriminant of the period equation," Amer. J. Math., v. 90, 1968, pp. 375379. MR 37 #2718. MR 0227133 (37:2718)
 [11]
 H. von LIENEN, "Primzahlen als achte Potenzreste," J. Reine Angew. Math., v. 266, 1974, pp. 107117. MR 49 #4916. MR 0340160 (49:4916)
 [12]
 G. B. MATHEWS, Theory of Numbers, 2nd ed., Chelsea, New York, 1961. MR 23 #A3698. MR 0126402 (23:A3698)
 [13]
 J. B. MUSKAT, "Criteria for solvability of certain congruences," Canad. J. Math., v. 16, 1964, pp. 343352. MR 29 #1170. MR 0163871 (29:1170)
 [14]
 J. B. MUSKAT, "On the solvability of ," Pacific J. Math., v. 14, 1964, pp. 257260. MR 28 #2997. MR 0159781 (28:2997)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197604120899
PII:
S 00255718(1976)04120899
Keywords:
Quintic residue,
primitive root,
fnomial periods,
period equation
Article copyright:
© Copyright 1976
American Mathematical Society
