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.
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 [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
