The 3x +1 Problem: An Annotated Bibliography (1963–1999) (Sorted by Author) Jeffrey C. Lagarias Department of Mathematics University of Michigan Ann Arbor, MI 48109–1109
[email protected] (January 1, 2011 version) ABSTRACT. The 3x + 1 problem concerns iteration of the map T : Z Z given by → 3x + 1 if x 1 (mod 2) . 2 ≡ T (x)= x if x 0 (mod 2) . 2 ≡ The 3x + 1 Conjecture asserts that each m 1 has some iterate T (k)(m) = 1. This 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. This version of the 3x + 1 bibliography sorts the papers alphabetically by the first author’s surname. A different version of this bibliography appears in the recent book: The Ultimate Challenge: The 3x + 1 Problem, Amer. Math. Soc, Providence, RI 2010. In it the papers are sorted by decade (1960-1969), (1970-1979), (1980-1989), (1990-1999). 1. Introduction The 3x + 1 problem is most simply stated in terms of the Collatz function C(x) defined on integers as “multiply by three and add one” for odd integers and “divide by two” for even integers. That is, arXiv:math/0309224v13 [math.NT] 11 Jan 2011 3x +1 if x 1 (mod 2) , ≡ C(x)= x if x 0 (mod 2) , 2 ≡ The 3x + 1 problem, or Collatz problem , is to prove that starting from any positive integer, some iterate of this function takes the value 1. Much work on the problem is stated in terms of the 3x + 1 function 3x + 1 if x 1 (mod 2) 2 ≡ T (x)= x if x 0 (mod 2) .