Lagarias J.C. "The 3x + 1 problem and its generalizations." Amer. Math Monthly 92, pp. 3--23. 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is undecidable. Proof. By application of Theorem 3.1. ut 3.2 Conway Unpredictable Iterations Here we present some work by J.H. Conway [3] which has studied a generalization of the Collatz conjecture. The exact origin of this conjecture also called Syracuse conjecture or 3x 1 problem [11] is not clearly known. It had circulated by word of mouth among the mathematical community for many years. This problem is credited to Lothar Collatz at the University of Hamburg. This conjecture asserts that the opposite program, given any integer n, always terminates. While n 1 Do If n is ....

Lagarias J.C. "The 3x + 1 problem and its generalizations." Amer. Math Monthly 92, pp. 3--23. 1985.

.... of Halbeisen and Hungerbuhler [1997] any extra cycle of positive integers must contain more than 10 8 numbers (even when counting any 3n 1 step and the subsequent division by 2 as one; see below) Surveys of the history and the variety of known results about the 3n 1 problem can be found in [Lagarias 1985; Wirsching 1998] Among the gems to be found there is a theorem of John H. Conway to the effect that a simple generalization of the 3n 1 problem is algorithmically undecidable because it encodes the halting problem for Turing machines, and a comment of Erdos saying that Mathematics is not yet ....

J. C. Lagarias, "The 3x + 1 problem and its generalizations", Amer. Math. Monthly 92:1 (1985), 3--23.

....The well known 3x 1 problem deals with the iterative behavior of the function T : N N (where N is the set of positive integers) defined as follows: T (n) ae n=2 if n is even, 3n 1) 2 if n is odd. All known numerical checks, as well as a few interesting heuristic arguments [Lagarias 1985], indicate that a typical trajectory (sequence of iterates) of T degenerates into repetitions of the finite cycle f1 2 1g. The 3x 1 conjecture asserts that this is true for any positive integer n. Since the problem became known about sixty years ago, many interesting and deep facts ....

....degenerates into repetitions of the finite cycle f1 2 1g. The 3x 1 conjecture asserts that this is true for any positive integer n. Since the problem became known about sixty years ago, many interesting and deep facts concerning the iteration of T have been discovered; most are reported in [Lagarias 1985], where one can find 70 relevant references. See also [Lagarias 1990; c fl A K Peters, Ltd. 1058 6458 1998 0.50 per page Experimental Mathematics 7:2, 146 Experimental Mathematics, Vol. 7 (1998) No. 2 Lagarias and Weiss 1992; Applegate and Lagarias 1995] Still, the 3x 1 conjecture remains ....

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J. C. Lagarias, "The 3x + 1 problem and its generalizations", Amer. Math. Monthly 92:1 (1985), 3--23.

....and the permutation of variables during inference generally make intuitive comprehension of behaviour impossible. The following example shows how difficult the problem of proving termination can be. The exact origin of the Collatz conjecture also called Syracuse conjecture or 3x 1 problem [Lag85] is not clearly known. It had circulated by word of mouth among the mathematical community for many years. This problem is credited to Lothar Collatz at the University of Hamburg. This conjecture asserts that the following program, given any integer n, always terminates. While n 1 Do If n is ....

Lagarias J.C. "The 3x + 1 problem and its generalizations." Amer. Math Monthly 92, pp. 3--23. 1985.

....one in one dimension based on counter machines that simulates Turing machines with an exponential slowdown, and another in two dimensions that simulates TMs in real time. Preliminary versions of these results appeared in [5] and [10] 2 One dimension: Minsky machines and Collatz functions Recall [6] the classic 3x 1 problem. If f is the function on the integers f(x) ae x=2 (x even) 3x 1 (x odd) then, for all x, does there exist a t such that f t (x) 1 In dynamical systems terms, is all of N in the basin of attraction of the periodic orbit f1; 4; 2g We can generalize this as ....

J.C. Lagarias, "The 3x + 1 problem and its generalizations." Amer. Math. Monthly 92 (1985) 3-23.

....n we call the minimal k such that T (k) n) 1 the total stopping time of n and denote it oe 1 (n) letting oe 1 (n) 1 if it is otherwise undefined. The 3x 1 function is a deterministic process that apparently exhibits pseudorandom behavior. It has been extensively studied; see the surveys [Lagarias 1985; Muller 1991] One approach to quantify its apparent pseudorandomness is to consider c fl A K Peters, Ltd. 1058 6458 96 0.50 per page 194 Experimental Mathematics, Vol. 4 (1995) No. 3 probabilistic models for its behavior on a random input, and then to compare model predictions with ....

....ff. One can rigorously prove that, when 0 ff 1, the successive iterates i log T (n) n ; log T [ffk] n) n j behave exactly like the trajectory of a random walk that takes i.i.d. independent, identically distributed) steps of size log 3 2 or log 1 2 with equal probability [Lagarias 1985, x 2] This suggests that the evolution of 3x 1 function iterates can be modelled by a multiplicative random walk, in which from an initial point X 0 one multiplies by successive i.i.d. random variables X i taking the values 3 2 and 1 2 with probability 1 2 each, to obtain Y j : X 0 X ....

J. C. Lagarias, "The 3x + 1 problem and its generalizations", Amer. Math. Monthly 92 (1985), 3--21.

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