The 3x+1 problem: An annotated bibliography (1963--1999)

The 3x+ 1 problem concerns iteration of the map on the integers given by T(n) = (3n+1)/2 if n is odd; T(n) = n/2 if n is even. The 3x+1 Conjecture asserts that for every positive integer n > 1 the forward orbit of n under iteration by T includes the integer 1. This paper is an annotated bibliography of work done on the 3x+1 problem and related problems from 1963 through 1999. At present the 3x+1 Conjecture remains unsolved.

[1]  R. Terras,et al.  A stopping time problem on the positive integers , 1976 .

[2]  Philippe Devienne,et al.  Halting Problem of One Binary Horn Clause is Undecidable , 1993, STACS.

[3]  R. Guy.  Unsolved Problems in Number Theory , 1981 .

[4]  Richard Dunn.  On Ulam's Problem , 1973 .

[5]  D. Boyd.  Which Rationals are Ratios of Pisot Sequences? , 1985, Canadian Mathematical Bulletin.

[6]  H. Müller.  Über eine klasse 2-adischer funktionen im zusammenhang mit dem „3X + 1“-problem , 1994 .

[7]  Yakov G. Sinai,et al.  Structure Theorem for (d, g, h)-Maps , 2002 .

[8]  Jeffrey C. Lagarias,et al.  Lower bounds for the total stopping time of 3x + 1 iterates , 2001, Math. Comput..

[9]  G. Wirsching.  Über das 3n + 1 Problem , 2000 .

[10]  Maurice Margenstern,et al.  Frontier between decidability and undecidability: a survey , 2000, Theor. Comput. Sci..

[11]  Kenneth G. Monks,et al.  The autoconjugacy of the 3x+1 function , 2004, Discret. Math..

[12]  Steven J. Miller,et al.  Benford's law, values of L-functions and the 3x+1 problem , 2004, math/0412003.

[13]  Frantisek Kascak.  Small Universal One-State Linear Operator Algorithm , 1992, MFCS.

[14]  S. Znám,et al.  A Note on the 3x + 1 Problem , 1987 .

[15]  K. Stolarsky.  A prelude to the 3x+1 problem , 1998 .

[16]  K. Matthews,et al.  A Markov approach to the generalized Syracuse algorithm , 1985 .

[17]  R. Guy.  John Isbell's Game of Beanstalk and John Conway's Game of Beans-Don't-Talk , 1986 .

[18]  K. Mahler,et al.  An unsolved problem on the powers of 3/2 , 1968, Journal of the Australian Mathematical Society.

[19]  Y. Sinai.  Uniform Distribution in the (3x+1)-Problem , 2003 .

[20]  I. Krasikov.  HOW MANY NUMBERS SATISFY THE 3X + 1 CONJECTURE? , 1989 .

[21]  M. Feix,et al.  Statistical properties of an iterated arithmetic mapping , 1994 .

[22]  R. Terras.  On the existence of a density , 1979 .

[23]  Jeffrey C. Lagarias,et al.  The 3x + 1 Problem and its Generalizations , 1985 .

[24]  B. D. Taylor,et al.  A new statistic for the 3x + 1 problem , 2001 .

[25]  Jeffrey C. Lagarias,et al.  THE 3x + 1 PROBLEM: TWO STOCHASTIC MODELS , 1992 .

[26]  Michael Avidon.  On Primitive 3-smooth Partitions of n , 1997, Electron. J. Comb..

[27]  Danut Marcu.  The powers of two and three , 1991, Discret. Math..

[28]  Guo-Gang Gao.  On consecutive numbers of the same height in the Collatz problem , 1993, Discret. Math..

[29]  E. Heppner.  Eine Bemerkung zum Hasse-Syracuse-Algorithmus , 1978 .

[30]  J. Sander.  On the (3N+1)-conjecture , 1990 .

[31]  G. Venturini.  On a Generalization of the 3 x+1 Problem , 1997 .

[32]  Clifford A. Reiter,et al.  Visualizing generalized 3x+1 function dynamics , 2001, Comput. Graph..

[33]  T. Brox.  Collatz cycles with few descents , 2000 .

[34]  D. Knuth,et al.  Recurrence relations based on minimization , 1974 .

[35]  Carl Pomerance,et al.  On a conjecture of Crandall concerning the qx + 1 problem , 1995 .

[36]  Games of cards, dynamical systems, and a characterization of the floor and ceiling functions , 1990 .

[37]  Jerzy Marcinkowski,et al.  Achilles, Turtle, and Undecidable Boundedness Problems for Small DATALOG Programs , 1999, SIAM J. Comput..

[38]  A. Gilman,et al.  A generalization of Everett's result on the Collatz 3x + 1 problem , 1987 .

[39]  L. Halbeisen,et al.  Optimal bounds for the length of rational Collatz cycles , 1997 .

[40]  Nicholas Pippenger.  An Elementary Approach to Some Analytic Asymptotics , 1993 .

[41]  Serge Burckel.  Functional Equations Associated with Congruential Functions , 1994, Theor. Comput. Sci..

[42]  Richard K. Guy,et al.  Conway's Prime Producing Machine , 1983 .

[43]  Tanguy Urvoy.  Regularity of Congruential Graphs , 2000, MFCS.

[44]  S. Brocco.  A Note On Mignosi′s Generalization of the (3X+1)-Problem , 1995 .

[45]  Daniel J. Bernstein,et al.  The 3x+1 conjugacy map , 1996 .

[46]  Lynn E. Garner.  On the Collatz $3n+1$ algorithm , 1981 .

[47]  G. Wirsching.  On the problem of positive predecessor density in $3n+1$ dynamics , 2003 .

[48]  G. Wirsching.  An improved estimate concerning 3n+1 predecessor sets , 1993 .

[49]  Richard E. Crandall,et al.  On the $‘‘3x+1”$ problem , 1978 .

[50]  Some Borel measures associated with the generalized Collatz mapping , 1992 .

[51]  M. García,et al.  A note on the generalized 3n + 1 problem , 1999 .

[52]  Arnold M. Zwicky,et al.  Three open questions in the theory of one-symbol Smullyan systems , 1970, SIGA.

[53]  David A. Klarner.  An Algorithm to Determine When Certain Sets Have 0-Density , 1981, J. Algorithms.

[54]  I. Korec.  The $3x+1$ problem, generalized Pascal triangles and cellular automata , 1992 .

[55]  Richard K. Guy,et al.  Don't Try to Solve These Problems! , 1983 .

[56]  Y. Matiyasevich,et al.  A binomial representation of the 3x + 1 problem , 1999 .

[57]  G. Venturini.  Iterates of Number Theoretic Functions with Periodic Rational Coefficients (Generalization of the 3x + 1 Problem) , 1992 .

[58]  Ivan Korec.  A density estimate for the $3x+1$ problem , 1994 .

[59]  Richard Rado,et al.  Arithmetic Properties of Certain Recursively Defined Sets. , 1974 .

[60]  J. Lagarias,et al.  Density bounds for the 3x + 1 problem. I: tree-search method , 1995 .

[61]  H. Möller.  Über Hasses Verallgemeinerung des Syracuse-Algorithmus (Kakutanis Problem) , 1978 .

[62]  Raymond Queneau.  Sur les suites s-additives , 1972 .

[63]  Stuart Anderson.  Struggling with the 3 x + 1 problem , 1987 .

[64]  Y. Sinai.  A Theorem About Uniform Distribution , 2004 .