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1998 Embedding the $3x+1$ conjecture in a $3x+d$ context
Edward G. Belaga, Maurice Mignotte
Experiment. Math. 7(2): 145-151 (1998).


Recall the well-known $3x+1$ conjecture: if $T(n)=(3n+1)$/$2$ for $n$ odd and $T(n)=n$/$2$ for $n$ even, repeated application of $T$ to any positive integer eventually leads to the cycle $$\{1\to2\to1\}\hbox{.}$$ We study a natural generalization of the function $T$, where instead of $3n+1$ one takes $3n+d$, for $d$ equal to -1 or to an odd positive integer not divisible by 3. With this generalization new cyclic phenomena appear, side by side with the general convergent dynamics typical of the $3x+1$ case. Nonetheless, experiments suggest the following conjecture: For any odd $d \ge -1$ not divisible by 3 there exists a finite set of positive integers such that iteration of the $3x+d$ function eventually lands in this set.

Along with a new boundedness result, we present here an improved formalism, more clear-cut and better suited for future experimental research.


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Edward G. Belaga. Maurice Mignotte. "Embedding the $3x+1$ conjecture in a $3x+d$ context." Experiment. Math. 7 (2) 145 - 151, 1998.


Published: 1998
First available in Project Euclid: 24 March 2003

zbMATH: 0918.11008
MathSciNet: MR1677087

Primary: 11B83
Secondary: 11K31

Keywords: $3x+1$ conjecture , $3x+1$ function , $3x+1$ problem , $3x+1$ trajectory , cycle , divergent trajectory , iteration of number-theoretic functions , termination set

Rights: Copyright © 1998 A K Peters, Ltd.


Vol.7 • No. 2 • 1998
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