Integer sequences having prescribed quadratic character
Authors:
D. H. Lehmer, Emma Lehmer and Daniel Shanks
Journal:
Math. Comp. 24 (1970), 433451
MSC:
Primary 10.03
MathSciNet review:
0271006
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: For the odd primes , we determine integer sequences such that the Legendre symbol for all for a prescribed array of signs ; (i.e., for a prescribed quadratic character). We examine six quadratic characters having special interest and applications. We present tables of these and examine some applications, particularly to questions concerning extreme values for the smallest primitive root (of a prime ), the class number of the quadratic field , the real Dirichlet functions, and quadratic character sums.
 [1]
W.
H. Mills, Characters with preassigned values, Canad. J. Math.
15 (1963), 169–171. MR 0156828
(28 #71)
 [2]
D. H. Lehmer, "An announcement concerning the Delay Line Sieve DLS127," Math. Comp., v. 20, 1966, pp. 645646.
 [3]
Marshall
Hall, Quadratic residues in
factorization, Bull. Amer. Math. Soc.
39 (1933), no. 10,
758–763. MR
1562725, http://dx.doi.org/10.1090/S000299041933057300
 [4]
Allan Cobham, The Recognition Problem for the Set of Perfect Squares, IBM Research Paper, R.C. 1704, April 26, 1966.
 [5]
M. Kraitchik, Recherches sur la Théorie des Nombres. Vol. 1, Paris, 1924, pp. 4146.
 [6]
D.
H. Lehmer, The Mechanical Combination of Linear Forms, Amer.
Math. Monthly 35 (1928), no. 3, 114–121. MR
1521394, http://dx.doi.org/10.2307/2299504
 [7]
D.
H. Lehmer, A sieve problem
on“pseudosquares.”, Math. Tables
and Other Aids to Computation 8 (1954), 241–242. MR 0063388
(16,113e), http://dx.doi.org/10.1090/S0025571819540063388X
 [8]
A.
E. Western and J.
C. P. Miller, Tables of indices and primitive roots, Royal
Society Mathematical Tables, Vol. 9, Published for the Royal Society at the
Cambridge University Press, London, 1968. MR 0246488
(39 #7792)
 [9]
Daniel
Shanks, Class number, a theory of factorization, and genera,
1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State
Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence,
R.I., 1971, pp. 415–440. MR 0316385
(47 #4932)
 [10]
N.
G. W. H. Beeger, Report on some calculations of prime numbers,
Nieuw Arch. Wiskde 20 (1939), 48–50. MR 0000393
(1,65g)
 [11]
Luigi Poletti, "Atlante di centomila numeri primi di ordine quadratico entro cinque miliardi," UMT 62, MTAC, v. 2, 1947, p. 354.
 [12]
H.
M. Stark, A complete determination of the complex quadratic fields
of classnumber one, Michigan Math. J. 14 (1967),
1–27. MR
0222050 (36 #5102)
 [13]
D. H. Lehmer, "On the function ," Sphinx, v. 6, 1936, pp. 212214; v. 7, 1937, p. 40; v. 9, 1939, pp. 8385.
 [14]
Mohan Lal & Daniel Shanks, "Class numbers and a high density of primes." (To appear.)
 [15]
R.
Ayoub, S.
Chowla, and H.
Walum, On sums involving quadratic characters, J. London Math.
Soc. 42 (1967), 152–154. MR 0204382
(34 #4224)
 [16]
Daniel
Shanks, Generalized Euler and class
numbers, Math. Comp. 21 (1967), 689–694. MR 0223295
(36 #6343), http://dx.doi.org/10.1090/S00255718196702232955
 [17]
Daniel
Shanks, On the conjecture of Hardy &
Littlewood concerning the number of primes of the form
𝑛²+𝑎, Math. Comp. 14 (1960), 320–332.
MR
0120203 (22 #10960), http://dx.doi.org/10.1090/S00255718196001202036
 [18]
Daniel
Shanks, Supplementary data and remarks
concerning a HardyLittlewood conjecture, Math.
Comp. 17 (1963),
188–193. MR 0159797
(28 #3013), http://dx.doi.org/10.1090/S00255718196301597976
 [19]
Daniel
Shanks, On Gauss’s class number
problems, Math. Comp. 23 (1969), 151–163. MR 0262204
(41 #6814), http://dx.doi.org/10.1090/S00255718196902622041
 [20]
Edward T. Ordman, "Tables of class numbers for negative prime discriminants," UMT 29, Math. Comp., v. 23, 1969, p. 458.
 [21]
Morris
Newman, Note on partitions modulo 5,
Math. Comp. 21 (1967), 481–482. MR 0227127
(37 #2712), http://dx.doi.org/10.1090/S00255718196702271270
 [22]
Peter Weinberger, Dissertation, University of California, Berkeley, Calif., June, 1969.
 [23]
Edgar
Karst, The congruence 2^{𝑝1}≡1
(𝑚𝑜𝑑 𝑝²) and quadratic forms with high
density of primes, Elem. Math. 22 (1967),
85–88. MR
0215777 (35 #6612)
 [24]
C. L. Siegel, "Über die Classenzahl quadratischer Zahlkorper," Acta Arith., v. 1, 1935, pp. 8386.
 [25]
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen. Bände 2, Chelsea, New York, 1953, §186, "Euler's Reihen," pp. 673676. MR 16, 904.
 [1]
 W. H. Mills, "Characters with preassigned values," Canad. J. Math., v. 15, 1962, pp. 169171. MR 28 #71. MR 0156828 (28:71)
 [2]
 D. H. Lehmer, "An announcement concerning the Delay Line Sieve DLS127," Math. Comp., v. 20, 1966, pp. 645646.
 [3]
 Marshall Hall, "Quadratic residues in factorization," Bull. Amer. Math. Soc., v. 39, 1933, pp. 758763. MR 1562725
 [4]
 Allan Cobham, The Recognition Problem for the Set of Perfect Squares, IBM Research Paper, R.C. 1704, April 26, 1966.
 [5]
 M. Kraitchik, Recherches sur la Théorie des Nombres. Vol. 1, Paris, 1924, pp. 4146.
 [6]
 D. H. Lehmer, "The mechanical combination of linear forms," Amer. Math. Monthly, v. 35, 1928, pp. 114121. MR 1521394
 [7]
 D. H. Lehmer, "A sieve problem on "pseudosquares"," MTAC, v. 8, 1954, pp. 241242. MR 16, 113. MR 0063388 (16:113e)
 [8]
 A. E. Western & J. C. P. Miller, Indices and Primitive Roots, Royal Soc. Math. Tables, v. 9, Cambridge Univ. Press, New York, 1968, p. xv. MR 0246488 (39:7792)
 [9]
 Daniel Shanks, "Class number, a theory of factorization, and genera." (To appear.) MR 0316385 (47:4932)
 [10]
 N. G. W. H. Beeger, "Report on some calculations of prime numbers," Nieuw. Arch. Wiskde, v. 20, 1939, pp. 4850. MR 1, 65. MR 0000393 (1:65g)
 [11]
 Luigi Poletti, "Atlante di centomila numeri primi di ordine quadratico entro cinque miliardi," UMT 62, MTAC, v. 2, 1947, p. 354.
 [12]
 H. M. Stark, "A complete determination of the complex quadratic fields of classnumber one," Michigan Math. J., v. 14, 1967, pp. 127. MR 36 #5102. MR 0222050 (36:5102)
 [13]
 D. H. Lehmer, "On the function ," Sphinx, v. 6, 1936, pp. 212214; v. 7, 1937, p. 40; v. 9, 1939, pp. 8385.
 [14]
 Mohan Lal & Daniel Shanks, "Class numbers and a high density of primes." (To appear.)
 [15]
 R. Ayoub, S. Chowla & H. Walum, "On sums involving quadratic characters," J. London Math. Soc., v. 42, 1967, pp. 152154. MR 34 #4224. MR 0204382 (34:4224)
 [16]
 Daniel Shanks, "Generalized Euler and class numbers," Math. Comp., v. 21, 1967, pp. 689694. MR 36 #6343. MR 0223295 (36:6343)
 [17]
 Daniel Shanks, "On the conjecture of Hardy & Littlewood concerning the number of primes of the form ," Math. Comp., v. 14, 1960, pp. 320332. MR 22 #10960. MR 0120203 (22:10960)
 [18]
 Daniel Shanks,."Supplementary data and remarks concerning a HardyLittlewood conjecture," Math. Comp., v. 17, 1963, pp. 188193. MR 28 #3013. MR 0159797 (28:3013)
 [19]
 Daniel Shanks, "On Gauss's class number problems," Math. Comp., v. 23, 1969, pp. 151163. MR 0262204 (41:6814)
 [20]
 Edward T. Ordman, "Tables of class numbers for negative prime discriminants," UMT 29, Math. Comp., v. 23, 1969, p. 458.
 [21]
 Morris Newman, "Table of the class number for prime, , ," UMT 50, Math. Comp., v. 23, 1969, p. 683. MR 0227127 (37:2712)
 [22]
 Peter Weinberger, Dissertation, University of California, Berkeley, Calif., June, 1969.
 [23]
 Edgar Karst, "The congruence and quadratic forms with high density of primes," Elem. Math., v. 22, 1967, pp. 8588. MR 35 #6612. MR 0215777 (35:6612)
 [24]
 C. L. Siegel, "Über die Classenzahl quadratischer Zahlkorper," Acta Arith., v. 1, 1935, pp. 8386.
 [25]
 E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen. Bände 2, Chelsea, New York, 1953, §186, "Euler's Reihen," pp. 673676. MR 16, 904.
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
10.03
Retrieve articles in all journals
with MSC:
10.03
Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819700271006X
PII:
S 00255718(1970)0271006X
Keywords:
Quadratic character,
sieves,
primitive roots,
class number,
Dirichlet functions,
quadratic character sums,
pseudosquares
Article copyright:
© Copyright 1970
American Mathematical Society
