Applications of a continued fraction algorithm to some class number problems
Author:
M. D. Hendy
Journal:
Math. Comp. 28 (1974), 267277
MSC:
Primary 12A50; Secondary 10F20, 12A25
MathSciNet review:
0330102
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Abstract: We make extensive use of Lagrange's algorithm for the evaluation of the quotients in the continued fraction expansion of the quadratic surd , where for and for . The recursively generated terms in his algorithm lead to all norms of primitive algebraic integers of less than , D being the discriminant. By ensuring that the values contain at most one small prime, we are able to generate sequences of determinants d of real quadratic fields whose genera usually contain more than one ideal class. Formulae for their fundamental units are given.
 [1]
G. Chrystal, Algebra. Part II, 2nd ed., A. and C. Black Ltd., London, 1931.
 [2]
E. L. Ince, Cycles of Reduced Ideals in Quadratic Fields, Mathematical Tables, vol. IV, British Association for the advancement of Science, London, 1934.
 [3]
K.
E. Kloss, Some numbertheoretic calculations, J. Res. Nat.
Bur. Standards Sect. B 69B (1965), 335–336. MR 0190057
(32 #7473)
 [4]
D.
H. Lehmer, Emma
Lehmer, and Daniel
Shanks, Integer sequences having prescribed
quadratic character, Math. Comp. 24 (1970), 433–451. MR 0271006
(42 #5889), http://dx.doi.org/10.1090/S0025571819700271006X
 [5]
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
 [6]
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
 [7]
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)
 [8]
Daniel Shanks, "An interesting sequence: ." (To appear.)
 [1]
 G. Chrystal, Algebra. Part II, 2nd ed., A. and C. Black Ltd., London, 1931.
 [2]
 E. L. Ince, Cycles of Reduced Ideals in Quadratic Fields, Mathematical Tables, vol. IV, British Association for the advancement of Science, London, 1934.
 [3]
 K. E. Kloss, "Some number theoretic calculations," J. Res. Nat. Bur. Standards Sect. B, v. 69B, 1965, pp. 335336. MR 32 #7473. MR 0190057 (32:7473)
 [4]
 D. H. Lehmer, Emma Lehmer & Daniel Shanks, "Integer sequences having prescribed quadratic character," Math. Comp., v. 24, 1970, pp. 433451. MR 42 #5889. MR 0271006 (42:5889)
 [5]
 Daniel Shanks, "On the conjecture of Hardy and Littlewood concerning the number of primes of the form ," Math. Comp., v. 14, 1960, pp. 320332. MR 22 #10960. MR 0120203 (22:10960)
 [6]
 Daniel Shanks, "On Gauss's class number problems," Math. Comp., v. 23, 1969, pp. 151163. MR 41 #6814. MR 0262204 (41:6814)
 [7]
 Daniel Shanks, "Class number, a theory of factorisation and genera," Proc. Sympos. Pure Math., vol. 20, Amer. Math. Soc., Providence, R.I., 1971, pp. 415440. MR 0316385 (47:4932)
 [8]
 Daniel Shanks, "An interesting sequence: ." (To appear.)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197403301022
PII:
S 00255718(1974)03301022
Keywords:
Principal ideals,
real quadratic field,
fundamental unit,
infinite continued fraction,
Lagrange algorithm,
class number,
genera,
Shanks sequence
Article copyright:
© Copyright 1974
American Mathematical Society
