On the problem

Author:
R. E. Crandall

Journal:
Math. Comp. **32** (1978), 1281-1292

MSC:
Primary 10A99

DOI:
https://doi.org/10.1090/S0025-5718-1978-0480321-3

MathSciNet review:
0480321

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Abstract | References | Similar Articles | Additional Information

Abstract: It is an open conjecture that for any positive odd integer *m* the function

*h*. Here we show that the number of which satisfy the conjecture is at least for a positive constant

*c*. A connection between the validity of the conjecture and the diophantine equation is established. It is shown that if the conjecture fails due to an occurrence , then

*k*is greater than 17985. Finally, an analogous "" problem is settled for certain pairs .

**[1]**J. H. CONWAY, "Unpredictable iterations,"*Proc.*1972*Number Theory Conference*, Univ. of Colorado, Boulder, Colorado, 1972, pp. 45-52. MR**0392904 (52:13717)****[2]**I. N. HERSTEIN & I. KAPLANSKY,*Matters Mathematical*, 1974.**[3]**C. J. EVERETT, "Iteration of the number-theoretic function ,"*Advances in Math.*, v. 25, 1977, pp.42-45. MR**0457344 (56:15552)****[4]**M. ABRAMOWITZ & I. STEGUN (Editors),*Handbook of Mathematical Functions*, 9th printing, Dover, New York, 1965.**[5]***Zentralblatt für Mathematik*, Band 233, p. 10041.**[6]**J. ROBERTS,*Elementary Number Theory--A Problem Oriented Approach*, M.I.T. Press, Cambridge, Mass., 1977. MR**0498337 (58:16472)****[7]**Y. A. KHINCHIN,*Continued Fractions*, Univ. of Chicago, Chicago, Ill., 1964. MR**0161833 (28:5037)****[8]***Pi Mu Epsilon Journal*, v. 5, 1972, pp. 338, 463.**[9]**S. S. PILLAI,*J. Indian Math. Soc.*, v. 19, 1931, pp. 1-11.**[10]**A. HERSCHEFELD,*Bull. Amer. Math. Soc.*, v. 42, 1936, pp. 231-234.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1978-0480321-3

Keywords:
Algorithm,
diophantine equation

Article copyright:
© Copyright 1978
American Mathematical Society