Where are all the prime numbers hidden?

What other numbers can be found mixed in with the prime numbers?

Is there a way to remove these other numbers in order to reveal the prime numbers?

I was able to answer all three questions over a three year period. Here are the answers.

The prime numbers can be found in the 12-note octave, specifically in columns 1 5 7 and 11. The columns are created by adding '12' in layers to each of these four bases indefinitely. All the prime numbers (except 2 and 3) can be found within these 4 columns.

The prime numbers are found mixed in with the composites that are created only by the prime numbers themselves. Composite numbers created by 2 are found in intervals 0 2 4 6 8 9 and 10 only. Composites created by 3 are found in intervals 0 3 6 and 9 only. Therefore 2 and 3 are not regular primes.

The method I found to remove the composites of prime numbers is shown in the text in pdf format I am uploading, and in the 40 second video which contains 13 images of this text posted on YouTube at this link:



I am no expert in prime numbers, but a beginner. My software skills are rudimentary. The algorithm I devised could contain basic flaws which escape me. I may not succeed in creating a software version. This is why I feel the need for dialogue, constructive criticism, and help.

 

Composite numbers created by 2 are found in intervals 0 2 4 6 8 and 10 only. (Removed 9.)

 

To the second question, I think you might like to read up on the Sieve of Eratosthenes. as a way of "smoking out" prime numbers. That can be Google'd.

 

I "discovered" the 1,3,7,9 property too in working on prime numbers. (I was also surprised, but should not have been to learn that (as a result of that property) that all primes >5 squared end in a 1 or a 9.

Have you noticed that for a given modulus that the ending digit is a repeating pattern?.

The Sieve of Eratosthenes that was recommended to you helps verify if a number is prime, but it won't help you generate composite numbers.

Once you identify any prime number any factors applied to it produces composite numbers. I played with mod 30 as a way of identifying a pattern.

Each prime pn then becomes” a "middle-factor" from the point of its square and onward creating a pattern from pn x pn, .which is repeated every 30 x pn in the table. See the following examples. (Note: I use the term "middle factor" to be any factors other than 1 and p.)


Figure 1 - Seven’s

This pattern is repeated every 210 (i.e., 30 x 7). The boxes from 217 show one repeat of the pattern.


Figure 2 - Eleven’s
This pattern is repeated every 330 (i.e., 30 x 11). The boxes from 341 show one repeat of the pattern. Note: Some composite numbers (e.g., 77) were already identified in figure 1 above.

I couldn't interpret your software code, but I do have a simulated BASIC program for generating primes.

PRINT “Enter N”; Input N
DIM A(N); A(1) = 1; X = 0; /* initialize */
FOR K = 3 to N; A(N) = 0; NEXT K
FOR K = 3 to N; Let d=0
IF (K > 2 and K/2 = integer) THEN d=1; ENDIF
IF (K > 3 and K/3 = integer) THEN d=1; ENDIF
IF (K > 5 and K/5 = integer) THEN d=1; ENDIF
IF d not=0 THEN let A(d)=1
NEXT K
FOR P = 2 to M; X=0
FOR K=2 to N; IF A(K)=0 AND X=0 THEN X=A(K)=K; NEXT K /* Find next prime */
FOR K=2 to N; M=K*X; IF M<N THEN A(M)=1; ENDIF; NEXT K /* Cross off */
NEXT P
FOR K = 2 to N; IF A(K) not= 1 Then PRINT A(K); NEXT K

It's not an actual program language as it stands - just a "meta-language".




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230302_reply_to_ddinunno_on_Math_Forums.txt

My interest with primes comes from an extension of work rewriting music theory based upon the 12-note octave rather than the 7-note Major scale. In other words, using base 12 as the foundation of music theory rather than base 7. Results are stunning, providing immediate musical literacy for all, but this is another topic entirely.

I first saw that all prime numbers, except 2 and 3, were contained in columns 1 5 7 and 11 when all the numbers were listed in 12 columns, from 0 to 11. This is not a proof.

I use 0 to 11 instead of 1 to 12 because all numbers divided by 12 give moduli 0 to 11. Also because the first note played in any melody has an interval of 0 semitones.

This is important because we can exclude all the numbers found in columns 0 2 3 4 6 8 9 10 when seeking to isolate the prime numbers. We can deal with only one third of the numbers rather than all of them. Three times less work and faster.

Someone on another forum alerted me to check base 6. When all the numbers are entered into these 6 columns, from 0 to 5, all the primes fall into columns 1 and 5. I assume but have no proof that all bases which are multiples of 6 also contain all the primes.

These base 6 multiples (6, 18, 24 etc) should also contain the same composite numbers as those found in base 12. No demonstration or proof yet.

I next found that these composites were all multiples of previous prime numbers, excluding 2 and 3, and starting with 5. I have no proof that this is the case, but assume this is so as I have not found any exceptions, nor expect any.

I did notice the pattern of the ending numbers in all four columns of base 12, 1 3 5 7 9. But some of these numbers were composites of previous primes. So ending numbers did not matter to me then.

If I look at the ending numbers of all the numbers in the four columns listed in a single series, you find the following pattern: (1 5 7 1 3 7 9 3 5 9) (1 5 7 1 3 7 9 3 5 9) ... Can this pattern help in finding primes? I don't know.

What was of interest to me was to find a way of creating all the multiples of the prime numbers starting with 5 in order to eliminate them from the list of all the primes and their composites, thus revealing all the primes.

This is what my method, which you could not interpret, actually does. I used multiplication, but you can also use addition.

I did program in Sinclair BASIC initially, which I enjoyed. I do not understand your simulated BASIC for now. I also do not understand your work with modulus 30.

The conclusion of my work is that to find all the primes, you must calculate all the composites of previous primes starting with 5, and eliminate those composites from the list of primes and of prime composites which base 12 provides.

I have not studied the Sieve of Eratosthenes, but expect it to be a way of finding and eliminating prime composites. It seems to me there can be no other way.

Thanks to the 12-note musical octave, my work produces a list of all the primes, and of all the prime composites created by using only primes 5 and above, containing exactly 1/3 of all possible numbers. It also produces a list of all composites that contain primes 2 and 3, containing exactly 2/3 of all possible numbers.

Thank you for your reply ddinunno.

 

1. It is true that most songs begin (and end) with the root note. (I enjoy writing piano songs - just simple melodies, but fun to do.)

2. The BASIC program just sets up an array and - just like the traditional approach of crossing off every 2nd then 3rd then 5th, etc. numbers it stores primes 1 to n in that array and then prints them.

3. If you get a chance to critique my attempt at proving Goldbach's Conjecture, I'd appreciate it.

3. The fact that every prime (> 3) can be expressed as 6d +/- 1, may help your ideas about base 6.

I developed the following proof, but there are others on the web.

Proof:
So, ‘d’ is any positive integer. When d=1, the results satisfying 6d +/- 1 are 5 and 7, which are both prime. There are positive integers that cannot be expressed in the form 6d +/- 1, and others that can be expressed as 6d +/- 1 that are not prime numbers. See Figure 3. The upper section shows example results while rows 1 through 4 show numbers that cannot be expressed in this form.



All positive integers greater than 3 are covered as the table is expanded … to the right. Column one has 4 through 9; column two has 10 through 15; column three has 16 through 21; etc.

Notice that ‘25’ is not a prime (it has a "middle factor" of 5), but it is in the form 6d+1 when d=4 (i.e., 25 = 6(4) + 1). More importantly, 6d is always even; 6d +/-1 is always odd; and 3 of the 4 Rows contain integers that are always even (So, no primes in rows 1, 2, or 3.).

Row 4 starts with 9 and increments by 6. So, every element has a middle-factor of 3 (i.e., no primes there either.) So, all primes must be in the 6d +/-1 rows (top two) above, QED.

 

Wait till you see the piano layout used to play melodies. I will post a picture as soon as I can later tonight.

 

I will study your post later. Here is the bitonal piano, with 6 red tones and 6 green tones per octave. Will be using white and grey tones on another keyboard. This layout needs testing in order to see its strengths and weaknesses in action.

 

ddinunno. when you wrote " I "discovered" the 1,3,7,9 property too in working on prime numbers." did you mean 1 5 7 11? Otherwise I don't know what you mean by 1,3,7,9.

I also don't understand your work with 'mod 30'. Maybe illustrating it with examples would help.

The initial algorithm can be better understood using a second shorter version which uses plain words rather than abbreviations.

Here is the file. I will produce a third version which I started if you do not understand the second version.

Then I will study your basic program for generating primes, and see if I can duplicate it.

It seems to me that f you want to calculate all the primes between any two numbers, you must know all the primes that exist from 5 on up to the prime that 'reaches' the first number when squared.

The bitonal layout pictured above is very powerful as the relationship between the colors of the notes in any series of notes (scales, chords, melodies) is identical for the six notes in one color, and identical but opposite in color for the six notes of the other color.

This is not the case with the standard layout, where different color patterns exist for the twelve keys, which multiplies the number of color patterns that you must remember for any chord or note series.

 

The 1,3,7,9 pattern is about the ones digit of prime numbers. All primes >5 end in 1,2,7, or 9. And, Integers ending in 1,3,7, or 9 multiplied by other integers ending in 1,3,7, or 9 produce only integers ending in 1,3,7, or 9. That's why squaring primes always end in 1 or 9 - because any combination of these digits squared ends in a 1 or 9.

Mod 30 pictures are useful. See the n= ending in 1,3,7,9's in this display of the lowest factor of primes. Thirty is the product of 2x3x5 - so these lowest factors repeat in 30's ... n=31 repeats n=1. (All higher primes fit in on the "blank" rows.



The "spirals" of primes begin to be displayed when these mod 30's are pictured as follows:

 

The idea behind the BASIC program is based on the "traditional method" of crossing off numbers x2, x3, x5, x7. etc.

When you cross off all 2's from 1 to n the lowest number is a 3; when you cross off every 3rd number, the lowest remaining number is a 5, etc. Each cross off - reveals the next prime! So, a program can produce primes from 1 to n by performing this crossing off and storing the primes revealed and getting the number for the next crossing off.

 

The method you are demonstrating is probably a highly condensed and abstract version of the original Sieve which involves all the numbers. I will find a textbook and study it in order to understand your explanations. Please provide link if you can. But this abstract version is not immediately understandable by a beginner student.

The approach I am using shows base 12 'magically' concentrating all primes 5 and above, and all their composites, in a single exclusive series. Initially I used addition. I shifted to multiplication in order to simplify matters, and was then able to complete my demonstration. It could probably be condensed by the same logic used in the official explanation.

This same approach could probably be applied to all the whole numbers as well, not only to those found in the exclusive series.

If you have read and understood the second version written in plain English, you should see how multiplication was systematized to reveal all the primes. And that even primary school students could understand how primes are revealed using this approach.

I assume my work is just a duplication of work initially done by mathematicians of previous generations, which was then condensed to into formulas, which were then fed into programs.

Once the primes are revealed using an approach that is understandable in a simple, concrete, and non-abstract way, it hopefully becomes much easier to analyze the different conjectures that abound concerning primes.

Do you know if the exclusive list of primes and prime composites starting with 5 and above was also well known to previous generations? If not, it should make finding primes three times easier.

 

https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes