Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Unexplained, confirmed reoccurring pattern in primes. Explanations?
Replies: 7   Last Post: Aug 10, 2013 1:01 AM

 Messages: [ Previous | Next ]
 Kenneth Lyon Posts: 2 From: Fort Smith, AR Registered: 5/13/12
Unexplained, confirmed reoccurring pattern in primes. Explanations?
Posted: May 13, 2012 10:52 PM
 First 5000 Chords of Primes in Binary.csv (145.6 K)

If you take all primes from 1 to 42 and also 25, adding 42n + this array produces a dataset that contains all possible prime numbers. I've confirmed this against the first several million known primes using scripts; no exceptions. Several numbers become irrelevant after the first series, so the relevant set is as follows with actual primes highlighted for this example:

(Tables won't post here, so I'm putting this in a CSV format. Hopefully the formatting won't be too butchered. If it is, paste in a spreadsheet and delimit by commas.)

Relevant Set: 1,5,11,13,17,19,23,25,29,31,37,41

n/base, 1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41
1 ,43, 47, 53, -55, 59, 61, -65, 67, 71, 73, 79, 83
2 ,-85, 89, -95, 97, 101, 103, 107, 109, 113,-115, -121, -125
3 ,127, 131, 137, 139,-143,-145, 149, 151,-155, 157, 163, 167
4 ,-169, 173, 179, 181,-185,-187, 191, 193, 197, 199, -205, -209
5 ,211, -215,-221, 223, 227, 229, 233,-235, 249, 241, -247, 251
(I've marked non-primes with a negative sign.)

This is curious for a number of reasons obviously. The main one being just why do prime numbers cycle around whole multiples of 42? Beyond that, there's also the compression capacity implied by this. Primes could be sequenced into binary strings using this approach, such as follows:

n/base:,1, 5,11,13,17,19,23,25,29,31,37,41,
1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1,
2, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0,
3, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1,
4, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0,
5, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1,

Using this, you can retrieve the primes by adding 42 times the row number plus the column's number.

Further, tie each row or "chord" (as I've liked to think of it) to two bytes and that leaves plenty of room for fully encompassing logic switches on the last 4 bits. By declaring (or secreting) a starting row n, a sequence of massive primes and dissonances could be passed in a few bytes. That strikes me as something useful to cryptography.

Additionally, there's the curiosity of the size of the range seeming to match musical chord structures. This last speculation I haven't yet found the means to test.

Regardless, this is something I discovered some years back and have never been able to find a single reference to. Just what is this phenomenon? Does anyone have any answers to this? If so, I'd love to hear them. Additionally if its something people want to discuss further, I'd much love that as well.

Best,
Kenneth Lyon - KLyon441@gmail.com

Date Subject Author
5/13/12 Kenneth Lyon
5/18/12 Daniel Saarimäki
5/19/12 Daniel Saarimäki
5/19/12 Daniel Saarimäki
5/20/12 Kenneth Lyon
5/21/12 Daniel Saarimäki
5/24/12 Daniel Saarimäki
8/10/13 Mark Rickert