Applications of a continued fraction algorithm to some class number problems

Author:
M. D. Hendy

Journal:
Math. Comp. **28** (1974), 267-277

MSC:
Primary 12A50; Secondary 10F20, 12A25

DOI:
https://doi.org/10.1090/S0025-5718-1974-0330102-2

MathSciNet review:
0330102

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Abstract | References | Similar Articles | Additional Information

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 number-theoretic calculations*, J. Res. Nat. Bur. Standards Sect. B**69B**(1965), 335–336. MR**0190057****[4]**D. H. Lehmer, Emma Lehmer, and Daniel Shanks,*Integer sequences having prescribed quadratic character*, Math. Comp.**24**(1970), 433–451. MR**0271006**, https://doi.org/10.1090/S0025-5718-1970-0271006-X**[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**, https://doi.org/10.1090/S0025-5718-1960-0120203-6**[6]**Daniel Shanks,*On Gauss’s class number problems*, Math. Comp.**23**(1969), 151–163. MR**0262204**, https://doi.org/10.1090/S0025-5718-1969-0262204-1**[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****[8]**Daniel Shanks, "An interesting sequence: ." (To appear.)

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0330102-2

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