Welcome to the resource topic for 2023/648
Title:
Collatz Computation Sequence for Sufficient Large Integers is Random
Authors: Wei Ren
Abstract:The main results in the paper are as follows: (1) We randomly select an
extremely large integer and verify whether it can return to 1. The
largest one has been verified has length of 6000000 bits, which is
overwhelmingly much larger than currently known and verified, e.g.,
128 bits, and its Collatz computation sequence consists of 28911397
I' and O’, only by an ordinary laptop. (2) We propose an dedicated
algorithm that can compute 3x+1 for extremely large integers in
million bit scale, by replacing multiplication with bit addition,
and further only by logical condition judgement. (3) We discovery
that the ratio - the count of O' over the count of I’ in
computation sequence goes to 1 asymptotically with the growth of
starting integers. (4) We further discover that once the length of
starting integer is sufficient large, e.g., 500000 bits, the
corresponding computation sequence (in which I' is replaced with 1 and O’ is replaced with 0), presents sufficient randomness as a bit
sequence. We firstly obtain the computation sequence of randomly
selected integer with L bit length, where L is 500000, 1000000,
2000000, 3000000, 4000000, 5000000, 6000000, by our proposed
algorithm for extremely large integers. We evaluate the randomness
of all computation sequences by both NIST SP 800-22 and GM/T
0005-2021. All sequences can pass the tests, and especially, the
larger the better. (5) We thus propose an algorithm for random bit
sequence generator by only using logical judgement (e.g., logic
gates) and less than 100 lines in ANSI C. The throughput of the
generator is about 625.693 bits/s over an ordinary laptop with Intel
Core i7 CPU (1.8GHz).
ePrint: https://eprint.iacr.org/2023/648
See all topics related to this paper.
Feel free to post resources that are related to this paper below.
Example resources include: implementations, explanation materials, talks, slides, links to previous discussions on other websites.
For more information, see the rules for Resource Topics .