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The \(3x + 1\) Problem and its Generalizations

by Jeffrey C. Lagarias

Year of Award: 1986

Award: Lester R. Ford

Publication Information: The American Mathematical Monthly, vol. 92, 1985, pp. 3-23


The "\(3x+1\)" problem, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, or Ulam's problem, concerns iterates of the function on the positive integers that maps an even number \(n\) to \(n/2\) and an odd number \(n\) to \(3n+1\). This paper surveys results known results and generalizations of this problem.

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About the Author: (from The American Mathematical Monthly, vol. 92 (1985)) Jeffrey C. Lagarias: I was first exposed to the \(3x + 1\) problem in 1967 as a high school student working at the National Bureau of Standards. Afterwards I worked on it from time to time. Out of curiosity and frustration I gradually became a historian of the problem, accumulating a collection of papers about it. This survey is a happy consequence. I obtained a PhD. (1974) in analytic number theory at MIT under the supervision of Harold Stark. I have been on the staff of AT&T Bell Laboratories since then, and have held visiting positions at the University of Maryland (mathematics) and Rutgers University (computer science). My research interests include computational complexity theory, number theory, and cryptography.


Subject classification(s): Algebra and Number Theory | Number Theory | Index
Publication Date: 
Wednesday, September 24, 2008