• ### EVOLUTIONARY MATHEMATICS AND SCIENCE FOR SOME OBSERVATIONS OF THE TERMINAL VALUE DISTRIBUTION IN THE PROCESS OF SUCCESSIVELY SUMMING UP ALL PRIME FACTORS OF A GIVEN NATURAL NUMBER

Author(s):
Hung-ping Tsao
Editor(s):
Lawrence K Wang (see profile)
Date:
2022
Group(s):
Science, Technology, Engineering, Arts and Mathematics (STEAM)
Subject(s):
Mathematics, Numbers, Prime, Factor tables
Item Type:
Book chapter
Tag(s):
terminal value
Permanent URL:
https://doi.org/10.17613/e8dh-n335
Abstract:
Tsao, Hung-ping (2022). Evolutionary Mathematics and Science for Some Observations of the Terminal Value Distribution in the Process of Successively Summing up all Prime Factors of a Given Natural Number. In: "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", Wang, Lawrence K. and Tsao, Hung-ping (editors). Volume 4, Number 8F, August 2022; 75 pages. Lenox Institute Press, MA, USA. No. STEAM-VOL4-NUM8F-AUG2022. ...............ABSTRACT: It is known that each natural number greater than 1 can be expressed as a unique product of prime numbers, namely the prime factorization. For example, 9=3×3 and 3+3=6, we can continue the same process to finally come up with 6=2×3 and 2+3=5, which is a prime number. Accordingly, we obtain 5 to be the terminal value of 9, denoted as t(9)=5. In relating to the above process of successively summing all prime factors of a given natural number, I made the following observations. Observation 1: Beyond 764, the frequency of 5 being the terminal number will become lesser and lesser than that of 7, 11 or 13 being the terminal number. On the other hand, beyond 1910, the probability of 5, 7, 11 or 13 being the terminal number will become lesser and lesser than 0.6. Observation 2: Within 1290, the probability of 5, 7, 11 or 13 being the terminal number is exactly equal to 0.6. Likewise for 1305, 1310, 1320, 1325, 1340, 1800, 1885, 1895, 1900, 1905 and 1910, but not beyond. Observation 3: Within 1325, the frequency of double terminal values (2 in a roll) from {5} is exactly the same as that from {7, 11, 13}. Likewise for 1476 and 1477, but not beyond. Observation 4: Within 1653 up to 1673, the frequency of triple terminal values (2 in a roll) from {5} is exactly the same as that from {7, 11, 13}, but not beyond.
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