## My rediscovery of The Sieve of Eratosthenes

oktober 18, 2021 Kommentarer inaktiverade för My rediscovery of The Sieve of Eratosthenes

I’ve been playing around with combining the heavenly stems and earthly branches (干支) in numerical form. The ten heavenly stems are represented by digits 0 – 9, while the earthly branches by the digits 0 – eleven, having created special digits for ten and eleven. (See following illustration). I arranged the combined digits in a circuit like a clock, and then colorized each according to how well they are divisable. See drawing underneath: I then noticed that the colors were symmetrical mirrored along the 0/60 – 30 line.

Then I saw that the primes were almost symmetrical too! And if I redifined these numbers as ”primes relative to 60”, they were also symmetrical. The definition being: ”those numbers that do not share the same prime factors with 60, i.e. 2, 3, 5”. This means that if x is a ”prime relative to 60” then there is also a y that is a ”prime relative to 60”, such that x+y = 60!

Actually I found out that this was basically because of the ”primes relative to 30 = 2x3x5” were also symmetrical, and that the symmetry would continue for each reiteration of digits. Thus, if x is prime relative to N, then N+x is too!

I even found this true with ”primes relative with 6 = 2×3” and with ”primes relative with 15 = 3×5″. I conjectured that this would be the case even for N = 210 = 2x3x5x7, which would eliminate those numbers divisable with 7.

I then found the following article on The Sieve of Eratosthenes: https://arxiv.org/pdf/1905.03117.pdf

I found out that ”primes relative to 30” are termed ”3-primes”.

I had independently discovered The Sieve of Eratosthenes and the results of the first two theorems in the article, even for a random pick of primes!

Here are two excerpts from the above article that illustrate this:  