• ### EVOLUTIONARY MATHEMATICS AND SCIENCE FOR ON THE CONJECTURE 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, Science--Study and teaching, Technology--Study and teaching
Item Type:
Book chapter
Tag(s):
Mathematical notation, Conjecture, terminal value, prime factorization, Mathematics Education, Science and technology studies (STS)
Permanent URL:
https://doi.org/10.17613/n0s4-k751
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, 84=2×2×3×7. By summing up all prime factors of 84, we obtain 2+2+3+7=14. Since 14=2×7, we further sum up all prime factors of 14 to obtain 2+7=9. Since 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 84, denoted as t(84)=5. In relating to the above process of successively summing all prime factors of a given natural number, I made the following conjecture in . Conjecture: Within a certain limit n, the probability of 5 being the terminal number is about the same as the probability of 7, 11 or 13 being the terminal number, which is approximately equal to 0.3. In this sequel of , I’ll point out that n_1=489 is the first limit for the above conjecture to be true, where the pertinent probability is 0.30470 and n_2=510 is the second limit with the pertinent probability being 0.30196. In both cases, the frequency of 5 being the terminal number is exactly the same as that of 7, 11 or 13 being the terminal number. However, in the case of n_3=1268, the frequency of the latter is slightly more than that of the former. New Conjecture: Beyond the third limit n_3=1268, 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, the probability of 5, 7, 11 or 13 being the terminal number will always be approximately equal to 0.6.
Notes:
KEYWORDS: Prime number, Prime factorization, Terminal value of the process of successively summing.
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