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Posted on 07/16/2025 12:48:23 PM PDT by Peter ODonnell
I wondered if any FRiends were interested in prime numbers. You'll know that prime numbers are integers, and those which have only two factors, 1 and themselves.
Except for 2, all other primes are odd numbers. By convention, the number 1 is not considered prime (but this is strictly a matter of definition because 1 does not entirely contradict the premise of being a multiple of 1 and itself).
I cannot post this entire thread in one post so if you come across it in an early stage, be aware that a number of posts will follow rapidly until I think the new system that I have developed is fully outlined.
Therefore I will ask anyone interested not to post comments or questions until tomorrow's date (July 17th) by which time I will have managed to post an entire overview of this interesting research.
Interest has always existed in systems to predict prime numbers, and not a lot of progress can be reported. But it occurred to me that if you could predict non-primes (basically the odd numbers that are not primes) then that would be equivalent to predicting primes. An odd number is either prime or non-prime.
2, 3, 5 and 7 are the four smallest primes. Their many multiples are of course all non-primes. The fact that 2 is prime allows us to remove all other even numbers from consideration. (someone once asked, is zero a prime? ... by several definitions it would not be, but also, the realm of prime numbers is defined to be positive integers, so for the same reason negative primes would form a separate field of inquiry).
Realizing that multiples of 3, 5 and 7 are relatively easy to diagnose among larger odd numbers, and multiples of 9 are also multiples of 3, I began to think of the problem as essentially a system to predict multiples of 11, 13 and all the other higher odd numbers. By the way, you can tell at a glance if a number is a multiple of 5, because it will end with 5. Numbers that are multiples of 3 also satisfy the requirement that adding their digits will produce a multiple of 3. So I can tell without doing any calculations that 1233321 is non-prime and a multiple of 3 because I see (12)(333)(21) and all components are multiples of 3. Even if the numbers are scrambled up, if you can reduce digits of an odd number to components that are multiples of 3, then that number is a multiple of 3. For example, 4973811 is a multiple of 3 -- digits add to 33, and number is a composite of (48)(117)(39). For multiples of seven unfortunately there is no easier way to detect those, than to divide by 7, although if it is composed of multiples of seven you'll know it is divisible by 7 (but unlike 3, it doesn't work to scramble up the order of valid components, 1421 is a multiple of 7 but 4121 is not).
In any case, I started to work on a matrix of odd numbers that excludes all multiples of 3, 5 and 7. This is quite easy to do because once you have the first set (1 to 209) all other sets are 210 higher successively (because 3 x 5 x 7 = 105, all other odd numbers can be listed in two columns as shown below, with the order being descending in the first column, and ascending in the second column.
01 209
11 199
13 197
17 193
19 191
23 187
29 181
31 179
37 173
41 169
43 167
47 163
53 157
59 151
61 149
67 143
71 139
73 137
79 131
83 127
89 121
97 113
101 109
103 107
It will be observed that there are 24 numbers in each column and the 11 and higher system operates with 48 of the first 105 odd numbers, which means that 57 are excluded (these are the 35 multiples of 3, plus the 21 multiples of 5, minus the 7 multiples of 15, plus the 15 multiples of 7, minus the 5 multiples of 21 and the 2 multiples of 35 not already excluded).
And every pair of columns from (A1-24, B1-24) out to infinity will be 210n higher in value. Just to illustrate, the third and fourth columns are added for illustration:
01 209 211 419
11 199 221 409
13 197 223 407
17 193 227 403
19 191 229 401
23 187 233 397
29 181 239 391
31 179 241 389
37 173 247 383
41 169 251 379
43 167 253 377
47 163 257 373
53 157 263 367
59 151 269 361
61 149 271 359
67 143 277 353
71 139 281 349
73 137 283 347
79 131 289 341
83 127 293 337
89 121 299 331
97 113 307 323
101 109 311 319
103 107 313 317
Now in all of these system numbers, all of the primes greater than 7 will be found, but not all the numbers in the matrix are primes. All of the multiples of 11, 13 and various higher primes, are contained also.
It was necessary to include 1 in the matrix because all numbers 1 + 210n are required, as they are either primes or derived non-primes. But with reference to column A (column designations will follow the widely used Excel column system) all numbers in it, aside from 1, are primes and since 2, 3, 5 and 7 are primes, there are actually 27 primes in the domain of column A. (24-1+4)
One parameter being studied as I develop the system is how many of the 24 numbers in each column are prime and non-prime. It is known that primes exist at a steadily diminishing rate up to infinity (there is no highest prime, but there is a highest discovered prime with many millions of digits). My work so far has reached into the 380k range where the average is 16 non-primes and 8 primes per column. It quickly expanded to nearly that frequency (15, 9) in the first 20,000 (200 columns, or 100 pairs of columns, contain all system members up to 20999, the first member of column 201 is 20101. In Excel format, column 201 is column GS). The frequency of primes does not steadily diminish along an even asymptotic curve but tends to jog up and down erratically, even out where I am now working, it's fairly common to find columns with 3 primes up to 12 in some instances. Various workers have been studying the rhythms of variation of prime frequency as a possible aid to prediction. I am holding to my opposite approach that predicting non-primes will prove to be easier and more subject possibly to determining equations. But I am nowhere near that point and I would imagine that this 11 and higher system is just a building block of a new avenue of investigation that could span several more generations of workers to achieve results. Or it could happen next week -- the entire point of the research is to discover predictable patterns.
It won't come until the system pushes out into the range of several millions that the frequency of prime numbers will drop again by the same amount (to 17, 7) and even into numbers with many dozens of digits the frequency of primes holds up well above 1 out of 24. Why does the frequency of primes drop off so slowly, after all, new primes are gobbling up members higher up in the system as each set moves through -- there are two reasons. First, each new prime can only capture numbers that are higher primes. When it comes to an already captured number (as 13 does at 143) it cannot add that already captured value to its total. And the range over which it operates continually expands. 13 takes (13/11) times as long to do the work that 11 does in knocking out potential primes as multiples. 17 takes (17/11) times as long, and loses more captures to both 11 and 13. This system of attrition means that by 41, a prime number is only a quarter as efficient as 11 at removing non-primes (in approximate terms, it is taking out half as many per range of multiples, and losing half of those as captures to lower primes). By primes in the 211-241 range, the new knockout rate is falling well below 10% of the smaller more effective numbers. Imagine how seldom a large prime number in the 300k range will manage to knock out a higher system member, but in fact, it will happen every time that high prime number multiplies a higher one. That product cannot be taken by any other combination.
So the first significant finding is that each prime builds up similar patterns of non-prime multiples. I call a non-prime a "capture" of the first prime factor it has. The first capture in the system is 121 (11 x 11). The next one is 143 (11 x 13). The first capture not made by 11 is 169 (13 x 13). There is a pattern to each set of captures, and it is a reversible mirror image over a span of 2p columns, where p is the prime generating the pattern. So for 11, the pattern is observable over 22 columns, or 11 pairs of columns, or within the domain 1 to 2309. There will be 47 multiples of 11 in that domain, formed by multiplying 11 by every number greater than 1 in columns A and B. Thereafter, there will be 48 multiples in every 22 columns, all appearing in the exact same sequence as the first 47 plus the cell A2 for 11, the prime generator (so cell W2 is a multiple of 11, as is cell AS2, BO2, CK2, DG2, EC2, EY2, FU2, GQ2 etc etc).
The pattern for multiples of 11 over 22 columns (including the placeholder for 11 itself which becomes valid in all other sets) is a mirror image with columns 11 and 12 (K and L) the reflecting mid-point. Here is how it appears, if I set up 22 columns and exclude all members except the 48 multiples of 11, then the pattern is as follows:
* [ * * * * * * * * * * * * * * * * * * ] *
[ * * * * * * * * * * * * * * * * * * * * ]
* * * [ * * * * * * * * * * * * * * ] * * *
* * * * * * * * * * [ ] * * * * * * * * * *
* * * * * * [ * * * * * * * * ] * * * * * *
* [ * * * * * * * * * * * * * * * * * * ] *
* * * * * * * * [ * * * * ] * * * * * * * *
* * * * [ * * * * * * * * * * * * ] * * * *
* * * * * * * [ * * * * * * ] * * * * * * *
* * * * * * [ * * * * * * * * ] * * * * * *
* * [ * * * * * * * * * * * * * * * * ] * *
* * * * * [ * * * * * * * * * * ] * * * * *
* * * * [ * * * * * * * * * * * * ] * * * *
* * * * * * * [ * * * * * * ] * * * * * * *
* * * * * * * * * * [ ] * * * * * * * * * *
* [ * * * * * * * * * * * * * * * * * * ] *
* * * * * * * * * [ * * ] * * * * * * * * *
* * * * * * * * [ * * * * ] * * * * * * * *
* * * [ * * * * * * * * * * * * * * ] * * *
* * * * * * * * * * [ ] * * * * * * * * * *
* [ * * * * * * * * * * * * * * * * * * ] *
* * * * [ * * * * * * * * * * * * ] * * * *
* * * [ * * * * * * * * * * * * * * ] * * *
* * * * * * * [ * * * * * * ] * * * * * * *
So if one inspects the pattern, it becomes apparent that it is a reflection, and multiples of 11 occupy each of the 24 rows on either side of the mid-point once only. The number that can exist in a column varies from 1 to 4 and the average is 24/11 or 2.18 approximately. As noted, in this first "panel" for 11, one member is still a prime, that being 11 itself. In every subsequent identical panel (in regard to multiples of 11) over each succeeding 22 columns, the same pattern will emerge. 11 will continue to extract potential primes and turn them into non-primes in exactly this same pattern to infinity.
For 13, there is a similar pattern but it spreads out over 26 columns, the central spine has three entries reflecting as well, but they are located in rows 1, 13, and 19 (instead of the 4, 15 and 20 for eleven). This larger spread continues into all subsequent columns and it takes two extra columns on either side of the "central reflection" for 13 to complete the pattern.
The central reflection for any given prime p is located at column p (not to be confused with Excel's column P) and column p+1. The first central reflection is created by multiples of 97, 101 and 103, together with 107, 109 and 113 ... but only for 11 and 13 do all three fall into the central columns. For 19 to 31, two sets are in the central columns and for 37 to 103, one set is found there. For numbers greater than 103, the central reflection is one or more columns apart. Once the whole system is "mapped" it becomes obvious that there is also an alignment at boundaries, for example with the eleven pattern, cells V2 and W2 are multiples of 11 (but W2 is the first member of the second panel). For primes greater than 210 and less than 420, the entire first panel is captured by lower primes so the first panel that can be studied is the second panel derived from larger primes, and it will be riddled with captures even so. The visual aspect of the ever-expanding patterns becomes more and more corrupted by the captures until it is not apparent to the observer even if distinctive codes are used for each member.
Now 13 loses an average of four potential captures per panel to 11, and 17 loses about seven in total to 11 and 13. This rate of capture and attrition slowly increases through the range and into the 380k portion of the system where I am now working away, it is quite rare to find a cell captured by the new arrivals in the system (613 squared is 375,769, and its first capture after this startup entry is 613 x 617 or 378,221). In that span, 11 has captured over sixty non-primes. I am cross-referencing my work with a list of primes with their ranks, available on kaggle and provided by Charles Averill. So far I have identified 30,000 primes and that is about 38% of the grid so far. I have looked at lists of primes in much higher ranges and found that the frequency drops off very slowly, so this system would continue to show a ratio like 20 non-primes to 4 primes well into the billions or even trillions. It has been proven that there can be no largest prime number, but at any given time there is a largest identified prime number.
Another pattern detected is that squares of primes only fall into a few rows. This is because the square of any odd number not ending in 5 has to end in 1 or 9. There are no odd numbers that have squares ending in 3 or 7. The square of a prime is always where it makes its first appearance as a capturing non-prime generator. As an aside one can note that all quads must end in 1 (because they are squares of numbers that end in 1 or 9). Thus if you are tortured in a foreign prison and asked on pain of death if 67893 is the fourth power of any integer, you can confidently say no (but they will then ask a more difficult question). You may also be interested to learn that no even numbers have squares ending in 2 or 8 (all end in 0, 4 or 6) and therefore fourth powers of even numbers can only end in zero or 6.
Every group of 13 panels for 11 will be 11 panels for 13, and this means that patterns of 11 and 13 captures repeat after 30,030. If a number (like 143) is a multiple of 11 and 13, captured by 11, then 143 + 30,030n will also have those attributes.
A "grand convergence" is where three primes all meet after their separate activity, for 11 x 13 x 17 the grand convergence is at 255,255. I mapped out what a much larger grand convergence would look like, and it's interesting that almost all numbers in many adjacent columns are taken up by a system of multiples of these participating members, but the four spaces below the 11-multiple contributions in rows 21 to 24 at the central reflection cannot be taken by any participant -- they can however fall to higher primes not in the grand convergence.
I think that eventually a study of this "11 and higher" non-prime generator matrix will reveal many other patterns and it may be possible to formulate equations that identify non-primes. Then the identification of primes will basically become an after-product of the identification of non-primes.
... I will see if anyone is interested enough to comment or ask questions, and go from there.
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The number of respondents to this post will be a prime number................
I find Prime Numbers interesting (in much the same way I find Pi interesting) but not really enough for me to ask any questions...
But I do know we have a number of math geeks here on FR, so you should have company shortly!
But is there a signal on the sub-harmonic ..Kenneth needs to know.
I have played around with prime numbers and formed the hypothesis that almost all prime numbers end in 3, 7 or 9. Not sure if it’s true.
It’s good to have a hobby.
What is the true meaning of Pi?
“...a multiple of 5, because it will end with 5...”
-
Or zero.
the ratio of a circle’s circumference to its diameter.
“...a multiple of 5, because it will end with 5...”
-
Or zero.
+++++++++++++++++++++++
I almost replied similarly, but the author is talking about odd multiples of 5...(I think)
I think there are equal numbers of primes ending in 1, 3, 7 and 9. There are only two exceptions, 2 and 5. The 48 row entries in two columns have equal numbers ending in 1, 3, 7 and 9; assuming that capture is equally distributed, that would knock out equal proportions in each row (which looks to be very close to being the case, it is probably fractionally different at any moment in time moving across the grid).
Note to all readers: Error in original post, first number in column 201 is 21001 (not 20101 as I typed). Wish we had an edit function. I know cowboyusa could use one.
Some might remember John from his stint at the National Review, which fired him for speaking the truth. From the Wikipedia article:
Derbyshire suggested that white and East Asian parents should talk to their children about the threats posed to their safety by black people. He also recommended that parents tell their children not to live in predominantly black communities. He included the line "If planning a trip to a beach or amusement park at some date, find out whether it is likely to be swamped with blacks on that date."Very sensible advice.
Well, I was following until we got about 1/4 the way in.
After which, my kids, who are math nerds would ting interest in this exercise.
It’s beyond me.
Interesting post. I suggest you read about the Sieve of Eratosthenes.
https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
"Predicting non-primes" and then eliminating them is what happens in the ancient algorithm called the Sieve of Eratosthenes.
Oh darn!
Thanks for posting.
I’ve been fascinated by prime numbers too, although I must admit it’s been at least 3 years since I was last thinking about them and possible ways to identify new primes quickly. I’m glad I’m not the only weirdo who loves primes. I’m convinced God gave us prime numbers for the great pleasure found in the challenge they pose. Your post had me digging through some of my old programs to refresh my memory on what I was striving for with them back in the day:
In those old (and sadly incomplete) musings, I was trying to develop more of an architectural solution, a way of pinpointing the location of primes in arrays without having to do any cumbersome numerical calculations, the goal being to quickly know their “geographical position” without the absolute necessity to calculate value unless needed.
Sadly I’m too old now to be intellectually helpful on the subject, as I just cant think deeply enough anymore, but its still all very much a fun mystery, and I’d enjoy perusing whatever thoughts you and others might have on the subject.
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