Density bounds for the problem. II. Krasikov inequalities

Authors:
David Applegate and Jeffrey C. Lagarias

Journal:
Math. Comp. **64** (1995), 427-438

MSC:
Primary 11B83; Secondary 11Y99

MathSciNet review:
1270613

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The function takes the values if *x* is odd and *x*/2 if *x* is even. Let *a* be any integer with . If counts the number of *n* with which eventually reach *a* under iteration by *T*, then for all sufficiently large *x*, . The proof is based on solving nonlinear programming problems constructed using difference inequalities of Krasikov.

**[1]**David Applegate and Jeffrey C. Lagarias,*Density bounds for the 3𝑥+1 problem. I. Tree-search method*, Math. Comp.**64**(1995), no. 209, 411–426. MR**1270612**, 10.1090/S0025-5718-1995-1270612-0**[2]**R. E. Crandall,*On the “3𝑥+1” problem*, Math. Comp.**32**(1978), no. 144, 1281–1292. MR**0480321**, 10.1090/S0025-5718-1978-0480321-3**[3]**I. Krasikov,*How many numbers satisfy the 3𝑋+1 conjecture?*, Internat. J. Math. Math. Sci.**12**(1989), no. 4, 791–796. MR**1024983**, 10.1155/S0161171289000979**[4]**Jeffrey C. Lagarias,*The 3𝑥+1 problem and its generalizations*, Amer. Math. Monthly**92**(1985), no. 1, 3–23. MR**777565**, 10.2307/2322189**[5]**J. W. Sander,*On the (3𝑁+1)-conjecture*, Acta Arith.**55**(1990), no. 3, 241–248. MR**1067972****[6]**Günther Wirsching,*An improved estimate concerning 3𝑛+1 predecessor sets*, Acta Arith.**63**(1993), no. 3, 205–210. MR**1218235**

Retrieve articles in *Mathematics of Computation*
with MSC:
11B83,
11Y99

Retrieve articles in all journals with MSC: 11B83, 11Y99

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1995-1270613-2

Article copyright:
© Copyright 1995
American Mathematical Society