New lower bounds for the size of a non-trivial loop in the Collatz 3x+1 and generalized px+q problem

Details New lower bounds for the size of a non-trivial loop in the Collatz 3x+1 and generalized px+q problem New lower bounds for the size of a non-trivial loop in the Collatz 3x+1 and generalized px+q problem

2009-07-17

arxiv.org/abs/0907.3086v4

Mathematics

General Mathematics

6 pages

Scientific paper

In the Collatz 3x+1 problem, there are 3 possibilities: Starting from any positive number, we either reach the trivial loop (1,4,2), end up in a non-trivial loop, or go until infinity. In this paper, we shall show that if a non-trivial loop with m odd numbers exists, then its minimum odd number is bounded above by a function of m. We shall also use that bound to calculate the least number of odd elements required for a non-trivial loop to exist. Also, the generalized bounds for the px+q problem are given.

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