*** March 2012 Update ***

This particular post on prime numbers seems to attract a fair bit of casual interest. It was written very early in my naive investigation of prime numbers (which remains fairly naive and perhaps a bit inept). People who stumble across this particular post might find some of the more recent work to be fairly interesting. Here are some of the more current posts:

k-tuples within primorials part 1

k-tuples within primorials part 2 (some corrections)

predicted k-tuples

more efficient erathosthenes sieve?

recursive pattern in developing the primes

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A Postulate about Patterns, Symmetry and Periodicity in

the Distribution of non-primes and Possible Primes

Pete Quinn, BGC Engineering Inc., Victoria, BC, Canada

John Quinn, Carleton University, Ottawa, ON, Canada

**ABSTRACT.** This paper suggests that recurring patterns of non-primes and possible primes exist for all primorials, P_{n}#, where P_{n} is the n^{th} prime. Further, it is suggested that the pattern of non-primes and possible primes is symmetric within P_{n}#. The process for determining the pattern of non-primes and possible primes is explained. These recurring symmetric patterns can be used to isolate all primes to P_{n}#, representing a significant advance on existing simple sieving techniques.

- INTRODUCTION

The distribution of prime numbers has been a topic of considerable interest to mathematicians throughout history. In recent years, the topic has gained particular practical importance, as modern public key encryption methods rely on use of very large prime numbers, and the difficulty in discovering them directly. Various methods are available to look for prime numbers, including simple sieving techniques such as the sieve of Erasthothenes. Prior research has identified and exploited patterns in non-prime distribution evident in small primorials, such as 6, where 6 = P_{2}# = 3#. Visualization efforts reveal that similar recurring patterns exist for larger primorials, and are postulated to exist for all primorials. Further, these recurring patterns are all symmetric within the primorial, and this symmetry can be used to advantage. This paper proposes a new sieving method that capitalizes on observed recurring symmetric patterns in the distribution of non-primes and possible primes within every primorial grouping.

- OBSERVED PATTERNS

All prime numbers other than 2 conform to the infinitely repeating distribution 1,0, where “1” implies a number may be prime (i.e. is a “possible prime”) and “0” implies that a number cannot be prime. This is the same as saying that, apart from the special case of 2, all even numbers are not prime, all prime numbers are odd, and, in the absence of further information, all odd numbers are candidates to be checked for primacy. This rule may be used to screen out potential candidates when looking for prime numbers. It is simple, easy to propagate forward, and easy to accept as a given. However, on its own it is not particularly helpful in isolating primes, particularly as P_{n} grows large, since it leave 50% of all numbers as possible primes to be checked or ruled out by other means.

It is similarly well known that a six number pattern, 1,0,0,0,1,0, possesses a similar recurrence to infinity; in every group of six numbers, the 2^{nd}, 3^{rd}, 4^{th} and 6^{th} are automatically excluded as primes, and the 1^{st} and 5^{th} represent the position of possible primes. All prime numbers must conform to this rule, so all prime numbers fall as the 1^{st} or 5^{th} numbers in a series of six; however, not all of the 1^{st} and 5^{th} numbers are prime.

The literature also includes references to patterns involving repetitions of 30 number sequences, or 1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0. A common thread between these three sets of repetitive patterns is that they are based on groups of numbers defined by the first three primorials, or products of successive prime numbers: P_{1}#, P_{2}# and P_{3}#, or 2, 6 and 30, where the primorial for P_{n} is defined as .

A second common thread is that all three patterns are symmetric about P_{n}#/2. For example, 1 is symmetric about 1 (P_{1}#/2), the 6 number pattern 1,0,0,0,1,0 is symmetric about 3, and the 30 number sequence is symmetric about 15.

A third common thread is that each pattern of non-primes and possible primes within recurring groups of P_{n}# successive integers is based on the P_{n} repetition of the P_{n-1} pattern (or P_{n-1} sieve, for convenience), followed by removal of all prime products of P_{n} between P_{n-1}# and P_{n}#. This will involve products of P_{n} with prime numbers between P_{n} and P_{n-1}#; however, one can take advantage of symmetry, obtain the prime products up to P_{n}#/2, and then reflect the result.

It is postulated here, but not proven, that these same essential common elements continue for all P_{n}. The following paragraphs and Figures illustrate the observed patterns for the first several values of P_{n}. The subsequent section illustrates how these patterns might be exploited for finding primes.

Figure 1 presents the distribution of the first 18 prime numbers within successive groups of six consecutive whole numbers. In this graph and all the similar graphs, the coordinates have been obtained by determining (N mod P_{n}#) for the x-value, and truncating (N/P_{n}# + 1) for the y-value.

Figure 2 shows the distribution of the first 100,000 prime numbers according to the same representation. Other than 2 and 3, the two primes involved in obtaining P_{2}#, the remaining primes all fall in two possible locations, which are symmetric about P_{2} (i.e. 3), 1 or 5. This simple pattern represents the very simple fact that, apart from 2 and 3, no prime may be even (a multiple or two) or a multiple of 3.

Figure 1. First 18 prime numbers distributed through (mod 6)

Figure 2. First 100,000 prime numbers distributed through (mod 6)

Figures 3 and 4 illustrate similar patterns for the distribution of prime numbers within groups of 30, P_{3}#. For comparison, the possible prime locations inferred from P_{2}#, or the P_{2}# sieve, are shown repeating across groups of 6, and actual primes up to 30 are shown along the bottom. One can see that the periodicity developed for P_{2}#, with primes possible at 1^{st} and 5^{th} positions with groups of 6, is preserved, but additional possibilities have been removed. The P_{2}# pattern, or P_{2}# sieve, can be used as a starting point to identify possible primes within P_{3}#. By careful inspection, one can determine that the newly removed possibilities, 5 and 25, represent the complete set of prime multiples of 5, P_{3}, between P_{3} and P_{3}#. Deleting these from the recurring P_{2}# sieve yields the full set of possible primes for P_{3}#, providing the P_{3}# sieve. In this unique case, removing the P_{3} prime products also removes all non-primes up to 30. For subsequent P_{n}#, it is necessary to remove prime products of primes higher than P_{n} to remove all non-primes to P_{n}#.

Figure 3. First 45 primes distributed through (mod 30)

Figure 4. First 100,000 primes distributed through (mod 30)

Figure 5 shows the first 100,000 primes distributed through (mod 210), or within groups of P_{4}#, or 7#. The (mod 30) sieve is also repeated across the top for comparison, along with actual primes up to 210 along the bottom, for comparison. All possible primes within repetitions of 210 fall within the possibilities suggested by the (mod 30) sieve; however, selected possibilities have again been removed. In this case, the following possibilities no longer exist: 7, 49, 77, 91, 119, 133, 161 and 203. These represent multiples of 7 by 1, 7, 11, 13, 17, 19, 23 and 29, or 1 plus all primes from P_{4} to P_{3}#.

By inspection, the distribution of possible primes remains symmetric about P_{4}#/2. Since the 7-fold repetition of the symmetric P_{3}# sieve must also be symmetric about P_{4}#/2, it follows that the possible primes removed to obtain the P_{4}# sieve must also be likewise symmetric.

The P_{4}# sieve, once developed as described above, may now be repeated P_{5} times as the first step in identifying the distribution of possible primes within repeating groups of P_{5}#, or 2310. First however, one could examine the distribution of actual prime numbers to 210, which in this case differs from the possible primes within P_{4}# groups, or the P_{4}# sieve. The following additional numbers have been removed from the P_{4}# sieve to leave only prime numbers to 210: 121, 143, 169 and 209. These are all products of prime numbers higher than 7, yielding products less than 210. Note there is no evident symmetry in the distribution of these numbers, nor is there symmetry in the distribution of prime numbers to 210. The obvious recurring symmetric patterns are limited to non-primes and possible primes.

Figure 5. First 100,000 primes distributed through (mod 210)

Similar distributions of prime numbers can be generated for any larger P_{n}# following the same general processes, yielding symmetric patterns of possible primes and non-primes that can be further vetted to leave only prime numbers through a simple generation of all possible combinations of prime products less than P_{n}# involving prime numbers larger than P_{n}.

Figure 6 shows the distribution of primes for P_{5}#, or 2310. A simple check reveals that the distribution of possible primes and non-primes remains symmetric. Obtaining the P_{5}# sieve from the 11-fold repetition of the P_{4}# sieve requires the removal of all prime products of 11. In this case, products of more than two primes are required, but for obtaining the P_{5}# sieve, all of these prime products must include 11 at least once. Recalling symmetry, this step can be simplified by only obtaining prime products of 11 up to 1155, and then reflecting the result about 1155.

The graphs for larger P_{n}# values are not presented, as the existing patterns are not visibly apparent at practical plotting scales; however, the same symmetry exists in the distribution of non-primes and possible primes, and the same simple processes can be used to obtain this distribution and to then obtain the primes up to P_{n}#. Note that in obtaining the primes to P_{n}#, it is accepted that the primes to P_{(n-1)}# have already been confirmed, and so the methodical process of removing additional prime products seeks to identify new primes from P_{(n-1)}# to P_{n}#.

Figure 6. First 100,000 primes distributed through (mod 2310)

The development of new patterns with increasing P_{n} follows predictable patterns as illustrated in Table 1. The number of possible primes grows with P_{n} following a predictable progression defined by the product of all values of (P_{n} -1) up to P_{n}. In other words, when a given primorial sieve for P_{n}# is repeated P_{(n+1)} times as the starting point for determining the P_{(n+1)}# sieve, a total of P_{(n+1)} unique prime products of P_{(n+1)} are eliminated to obtain the higher sieve.

Ordinal, n |
n^{th} prime, P_{n} |
n^{th} primorial, P_{n}# |
Number of primes to P_{n}# |
Largest prime to P_{n}# |
Possible Primes from P_{n}# sieve |

1 | 2 | 2 | 1 | 2 | 1 |

2 | 3 | 6 | 3 | 5 | 2 |

3 | 5 | 30 | 10 | 29 | 8 |

4 | 7 | 210 | 46 | 199 | 48 |

5 | 11 | 2310 | 342 | 2309 | 480 |

6 | 13 | 30030 | 3248 | 30029 | 5760 |

7 | 17 | 510510 | 42331 | 510481 | 92160 |

Pattern: |

Table 1. Growth of patterns of non-primes and possible primes

- EXPLOITING THE OBSERVED PATTERNS – FINDING PRIME NUMBERS

It should now be evident that the process for isolating prime numbers on the basis of recurring symmetric patterns within groups of successive numbers P_{n}# long is straightforward, as has been explained for P_{n}# up to 2310 in the preceding section. To obtain the primes for some arbitrary larger P_{n}#, it is necessary first to know all primes up to P_{(n-1)}#. It would be convenient to also have the P_{(n-1)}# sieve, but not necessary as this can be obtained from the primes to P_{(n-1)}#.

To obtain the P_{(n-1)}# sieve, take the list of primes to P_{(n-1)}# and collapse it onto itself P_{(n-1)} times to a single list of possibilities within P_{(n-2)}#, from which we can directly infer the P_{(n-2)}# sieve. Recall now that the process for obtaining the P_{(n-1)}# sieve is to simply repeat the smaller sieve P_{(n-1)} times, then remove all prime products of P_{(n-1)} that fall between P_{(n-2)}# and P_{(n-1)}#. Recall again that we only need remove the prime products to P_{(n-1)}#/2, and can then reflect these about P_{(n-1)}#/2 to obtain the complete list of relevant prime products of P_{(n-1)}.

With the P_{(n-1)}# sieve established, we now generate the P_{n}# sieve in the same fashion, first repeating the P_{(n-1)}# sieve P_{n} times, then removing all prime products of P_{n}. The final step to obtain the primes to P_{n}# is to remove the additional prime products of primes greater than P_{n} less than P_{n}#.

- CONCLUSIONS

This paper has described the existence of recurring symmetric patterns in the distribution of non-primes and possible primes based on primorials. The observed patterns have been used to develop a simple technique for isolating all prime numbers up to a given primorial value.

As P_{n}# becomes arbitrarily large, this process becomes progressively more cumbersome; however, it is believed to represent a substantial simplification of more basic sieving techniques such as the sieve of Erathstothenes. The potential differences in computational complexity are beyond the scope of this paper, but may be of interest to others to check. It is believed that a mathematical description of the described recurring symmetric patterns, which is left for others, may lead to other simplifications of existing techniques for identifying very large primes.

- ACKNOWLEDGEMENTS

This work was first inspired by a lecture on cryptography given during a visit to the University of Waterloo, attended by the second author, resulting in an interest in the distribution of primes. The first author developed an interest after stumbling across Ulam’s Spiral in *The Math Book* by Clifford Pickover. The patterns were discovered by accident when programming Ulam’s spiral in a spreadsheet proved more than a few minute job, so instead the primes were plotted in successive circles, with 360 possibilities for each circle. Since 360 is a multiple of a primorial, 30, a beautiful pattern appeared, leading to a search for further patterns.

REFERENCES

Ayres and Castro, 2005, Hidden Structure in the randomness of Prime Number sequence?

Cattani, 2010, Fractal Patterns in the Prime Number Distribution

Gibbs, 2008, The Double-Helix Pattern of Prime Number Growth

Grainville, 2008, Prime Number Patterns

Pickover, 2010?, The Math Book

Hi Pete,

I just left you a note on Tao#s blog, but thought it would be quicker to leave it here:

What you’re saying about prime patterns seems basically correct to me.

See my blog for a related attempt to prove the twin prime conjecture, which relies on symmetrical primorial patterns..http://barkerhugh.blogspot.com/

Thanks Hugh, appreciate the input. I’ll have a look through your blog when I get some free time and see if I can follow it.

Cheers,

Pete

Hi Pete,

One quick thing to think about – the patterns you are looking at can be seen in a purer form if you isolate either the 6n-1 or 6n+1 number lines. Within these you can see an identical pattern of 5-multiples and 7-multiples running from 35 to 245 and 175 to 385.

This basic repeating pattern does create a symmetrical pattern in the combined number lines, but it is more clear that it is a fractal pattern if you look at them separately.

Cheers

Hugh, I’m not sure I follow…?

I mean take all numbers that can be expressed as 6n-1 – 5,11,17 etc

Then look at the pattern of 5-multiples and 7-multiples within that line.

3541 47 53 5965717783 8995101 107 113119 125131 137 143149

155 161167 173 179185191 197203209215221 227 233 239245(crosses fingers that formatting looks OK…)

This is a much purer symmetry (and more obviously fractal) than the overall pattern from 0 to 210 because there is one 5-multiple every 5 spaces, one 7-multiple every 7 spaces…

This pattern repeats from 175 to 285 in the 6n+1 number line (7, 13, 19…)

What you see from 0 to 210 is a symmetrical pattern, but it kind of lays these two patterns on top of each other – I think it’s helpful to see them separately.

On my blog, the last post I made about primorial fractal patterns explains this at more length.

OK thanks Hugh, I will try to look at what you are talking about. Likely won’t be today, but maybe if I can find a few free minutes.

Cheers,

Pete

Oh, I meant 175 to 385 in the 6n+1 line…

Hi,

This is satish kumar singh.I am working in field of prime number.My thinking or way of approach is similar to yours.I want a demonstration of it .Will u suggest me the way.

thanks

satish

http://satishprimenumber.blogspot.com

09011056492

Hi Satish, sorry for the tardy reply, I haven’t checked this in a while, and have been absorbed by work.

I’m not sure exactly what you’re asking me for, as far as a suggestion is concerned, nor am I sure I can help. But can you maybe explain your question a bit more completely?

Cheers,

Pete

I am writing book Distribution of prime and composite number.U can understand it batter.So i would like to take a suggestion of u r on it.

Please look at http://www.ma.utexas.edu/mp_arc/c/06/06-314.pdf

Thanks for the link, I had found that paper several months ago.

Cheers,

Pete

Pete,

You might be interested in this paper to see the ratio of growth stabilize between the two poles:

http://www.wseas.org/multimedia/journals/mathematics/2013/55-223.pdf

Ernie

Thanks for the note Ernie, and the link to the paper. I will read it when I find some time.

Best,

Pete

Okay, hope you enjoy it. By the way, I think it’s Hibbs (not Gibbs) for 2008, The Double-Helix Pattern of Prime Number Growth.

Take care,

Ernie

My question is that, is it any pseudo pattern gives prime number in proper range? If some set of number can give you each and every prime number in fixed range? Range is like this

2 to 4

3 to 9

5 to 25

11 to 121

13 to 169

.

.

.

n to n*n

I have way to find out prime number. It is simple. Most important things is that its not a time taking. Just i will have to subtract a unique set of number from fixed number.

example no 1 ( This give us prime no grater then equal to 11 and less then equal to 121).

210-199=11

210-197=13

210-193=17

210-191=19

210-187=23

210-181=29

210-179=31

210-173=37

210-169=41

210-167=43

210-163=47

210-157=53

210-151=59

210-149=61

210-143=67

210-139=71

210-137=73

210-131=79

210-127=83

210-121=89

210-113=97

210-109=101

210-107=103

210-103=107

210-101=109

210- 97=113

I can find out any prime number.

I have used unique set of number and subtract it from fixed number as in example no 1.

example no 2.( This give us prime no grater then equal to 13 and less then equal to 169)

2310-2297=13

2310-2293=17

2310-2291=19

2310-2287=23

2310-2281=29

2310-2279=31

2310-2273=37

2310-2269=41

2310-2267=43

2310-2263=47

2310-2257=53

2310-2251=59

2310-2249=61

2310-2243=67

2310-2239=71

2310-2237=73

2310-2231=79

2310-2227=83

2310-2221=89

2310-2213=97

2310-2209=101

2310-2207=103

2310-2203=107

2310-2201=109

2310-2197=113

2310-2183=127

2310-2179=131

2310-2173=137

2310-2171=139

2310-2161=149

2310-2159=151

2310-2153=157

2310-2147=163

2310-2143=167.

Sir I can find any prime number of multi digits. But calculating limit of computer do not allow me to work further. So please help me in this regards.

Sir i had already solved so many things in prime number representation is required.

So my request is if possible plz suggest me the platform where i can represent it.

Prime No Even N0 Prime No-Even No=Prime No

427 420 7

431 420 11

433 420 13

437 420 17

439 420 19

443 420 23

449 420 29

451 420 31

457 420 37

461 420 41

463 420 43

467 420 47

Prime No Even N0 Prime No-Even No=Prime No

457 450 7

461 450 11

463 450 13

467 450 17

469 450 19

473 450 23

479 450 29

481 450 31

487 450 37

491 450 41

493 450 43

497 450 47

Prime No Even N0 Prime No-Even No=Prime No

487 480 7

491 480 11

493 480 13

497 480 17

499 480 19

503 480 23

509 480 29

511 480 31

517 480 37

521 480 41

523 480 43

527 480 47

Prime No Even N0 Prime No-Even No=Prime No

517 510 7

521 510 11

523 510 13

527 510 17

529 510 19

533 510 23

539 510 29

541 510 31

547 510 37

551 510 41

553 510 43

557 510 47

Same way

Can we find out prime number same fashion

Prime NO Even no Prime NO-Even no= Prime No

5 2 3

7 2 5

11 6 5

13 6 7

17 6 11

19 6 13

23 6 17

29 6 23

37 30 7

41 30 11

43 30 13

47 30 17

53 30 23

59 30 29

61 30 31

67 30 37

71 30 41

73 30 43

223 210 13

227 210 17

229 210 19

233 210 23

239 210 29

241 210 31

251 210 41

257 210 47

263 210 53

269 210 59

271 210 61

277 210 67

281 210 71

283 210 73

293 210 83

307 210 97

311 210 101

313 210 103

317 210 107

2333 2310 23

2339 2310 29

2341 2310 31

2347 2310 37

2351 2310 41

2357 2310 47

2371 2310 61

2377 2310 67

2381 2310 71

2383 2310 73

2389 2310 79

2393 2310 83

2399 2310 89

2411 2310 101

2417 2310 107

2423 2310 113

2437 2310 127

2441 2310 131

2447 2310 137

2459 2310 149

2467 2310 157

2473 2310 163

2477 2310 167

30047 30030 17

30059 30030 29

30071 30030 41

30089 30030 59

30091 30030 61

30097 30030 67

30103 30030 73

30109 30030 79

30113 30030 83

30119 30030 89

30133 30030 103

30137 30030 107

30139 30030 109

30161 30030 131

30169 30030 139

30181 30030 151

30187 30030 157

30197 30030 167

30203 30030 173

30211 30030 181

30223 30030 193

30241 30030 211

30253 30030 223

30259 30030 229

30269 30030 239

30271 30030 241

30293 30030 263

30307 30030 277

30313 30030 283

30319 30030 289

Br

Satish kumar singh

9681572800