Originally Posted by

**Harmonic**
Hey there Anthony,

Welcome to our forum!

I looked at your analysis of prime numbers modulo 9 (the second link). You said: The first thing I did was to simply apply Mod 9 to the first 65000 Prime numbers, only to discover that the results exclusively returned 1 2 4 5 7 or 8 for numbers greater than or equal to 5. I could find this information nowhere else on the Internet.

Please note that the 72 number sequence that I discovered in Music, being 6 octaves of 12, led me to arrange the numbers 1 to 72 and place them into 6 columns as laid out above. This layout quickly shows why prime numbers will exclusively return these remainder values infinitely because the Prime numbers only occur in Column 1 and Column 5 which simply repeat 7 4 1 and 5 2 8 respectively.

That patterns you found are common knowledge and really quite trivial. There was no need to apply mod 9 to all those numbers because the result can be demonstrated quite easily for all numbers. The reason the primes mod 9 yield a residue in the set (1,2,4,5,7,8) is because no prime can be a multiple of 3 and 9 = 3 x 3. That's why 3, 6, and 9 are missing from the set of residues - the are not relatively prime to 9 whereas the other numbers are. Likewise, the reason the primes appear only in columns 1 and 5 when you list all natural numbers in six columns is because all primes greater than 3 are of the forum 6n +/- 1. This is because 6n + k is composite for k = (0,2,3,4) as should be obvious. Thus, they must be either in the first or the fifth column.

Likewise, the reason you get a residue of (1,4,7) in the first column and (2,5,8) in the second is because the primes must also be of the form 3n +/- 1 since otherwise they'd be be multiples of three. Specifically: 1 = 0 + 1, 4 = 3 + 1, 7 = 6 + 1, and 2 = 3 - 1, 5 = 6 - 1, and 8 = 9 - 1. It's all elementary number theory.

Mathematics can be a lot of fun, but the results can be quite misleading if you are not trained in the fundamental theory.

All the best,

Richard

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