Lower bounds for the total stopping time of 3X+1 iterates

Details Lower bounds for the total stopping time of 3X+1 iterates Lower bounds for the total stopping time of 3X+1 iterates

2001-03-08

arxiv.org/abs/math/0103054v1

Math. Comp. 72 (2003), 1035--1049.

Mathematics

Number Theory

21 pages latex, 1 figure

Scientific paper

The 3X+1 function T(n) is (3n+1)/2 if n is odd and n/2 if n is even. The total stopping time \sigma_\infty (n) for a positive integer n is the number of iterations of the 3x+1 function to reach 1 starting from n, and is \infty if 1 is never reached. The 3x+1 conjecture states that this function is finite. We show that infinitely many n have a finite total stopping time with \sigma_\infty(n) > 6.14316 log n. The proof uses a very large computation. It is believed that almost all positive integers have \sigma_\infty (n) > 6.95212 \log n. The method of the paper should extend to prove infinitely many integers have this property, but it would require a much larger computation.

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