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Research Collection

Working Paper

On the structure of Primes

Author(s):

Mansour, Mohamed Publication Date:

2020-10

Permanent Link:

https://doi.org/10.3929/ethz-b-000447009

Rights / License:

In Copyright - Non-Commercial Use Permitted

This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use.

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ON THE STRUCTURE OF PRIMES

M. Mansour Institute of Automatic Control

ETH Zurich. Oct. 2020 ABSTRACT

Composites are members of an infinite number of infinite arithmetic series. It is shown here that primes are members of an infinite number of finite linear arithmetic series. Is is shown here for a range of small primes. For very large range of primes there is no reason for this to be not valid.

1-INTRODUCTION

The Composites are given by x2+2 x k

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where x is odd number 3,5,7,9. . . and k=0— ∞ . Some series are included in others, e.g. x=9 is included in x=3 . If the primes p are known then (1) can be written:

p2+2 p k (2)

where p = 3 ,5 ,7,11,. . . .. and k= 0. . ... ∞.

These are infinite number of infinite arithmetic series.

Definition 1: A linear arithmetic series has steps which are increasing linearly, e.g. the step equals 2k ,k=0, 1, 2,. . .

5 ,7, 11, 17, 25, 35, 47, 61. . .is a linear arithmetic series with steps 2 ,4 ,6 ,8 ,10 ,12 ,14. . . . . .

Generally the series is given by a +P

bk where k=0 till k=0. . ..c (3)

. In this paper b= 2 or b= 6 and a is ≥5.

For a decreasing series we have a -P bk Here we have a finite linear arithmetic series.

Definition 2: The base for the series are the even values between the steps and 0. The base gives the even values which if added to the prime give the sequence of the series P

2k , k=

1 till k=1,2,3,. . . .. or P

6k , k= 1 till k= 1,2,3. . . .

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Base for b=2:

2 6 12 20 30 42 56 72 90 110 132 156 182 210 240 272 306 342 380 420 462 506 650 702 756 812 870. . . .

Base for b=6:

6 18 36 60 90 126 168 216 270 330 396 468 546

630 720 816 918 1026 1140. . . .

2-DESCRIPTION OF PRIMES OTHER THAN 2 ,3

1) A complete increasing linear finite arithmetic series of primes begins with p and ends with p2 .

All are primes exceptp2 p + P

2k from k=0 till k = 0. . . .p-2 Examples:. 5+ P

2k , 11 + 2k , 17+ 2k , 41+ 2k. The last one was found by Euler

2) A complete decreasing linear finite arithmetic series of primes begins withp2 and ends with p

Examples 25 -P

2k , 49 -P

2k , 169 - P

2k , k=1 till k=1. . .p-1 109 - P

2k ,k=0 till k =0. . . 9 is a special incomplete series. This prime series is : 109 ,107 ,103 ,97 ,89, 79, 67 ,53 ,37 ,19

It begins with a prime and ends with the least possible prime

3) An incomplete increasing linear arithmetic series of primes begins with p and the first step is even and >2

p + P

2k k= a till k= a. . . c other than the starting prime p Examples:

5 + P

2k k=3 till k= 3. . .6 5 11 19 29 41 notice that the next number in the series 55 is composite

7 + P

2k k=8 till k=8. . . 12 7 23 41 61 83 107 notice that the next number in the series 133 is composite

29 +P

2k k=34 till k=34. . .39 29 97 167 239 313 389 467 547 notice that the next number in the series 629 is composite

There are infinite number of incomplete series.

Remark 1: For left primes there is no complete increasing series because the first step leads to composites with factor 3 ( see the base)

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Remark 2: For primes ending with 9 there is no complete increasing series because the second step leads to composites with factor 5.

Also there is no decreasing complete series because the square ends with 1 so that the second step leads to composites with factor 5

Remark 3: For primes ending with 3 there is no complete increasing series because the first step leads to composites with factor 5

Remark 4: For primes ending with 1 there is no complete series other than 11 and 41.

Remark 5: for primes ending with 7 there is no complete increasing series other than 17.

Base: 2 6 12 20 30 42 56 72 90 110 132 156 182 210 240 272 306 342 380 420 462 506 650 702 756 812 870. . . .

4) If we consider the concept of right and left primes (right primes are primes in 5+6k and left primes are primes in 7+6k) then we have the

Following:

1) A complete right series begins with p and ends with p(3p-2). All are primes except the last number

p + P

6k , k=0 till k=0. . .p-2 Examples:

5 11 23 41 the next number (65) is composite 5 x 13 .Decreasing complete series:

(65) 59 47 29 5

11 17 29 47 71 101 137 179 227 281 the next number is 341= 11 x 31

23 29 41 59 83 113 149 191 239 293 353 419 491 569 653 743 839 941 1049 1163 1283 1409 the next number is

1541 = 23 x 67

With similar arguments as in the b= 2 case we see that there are no increasingcomplete series other than these three.

65 -P

6k , k=1 till k=1. . .4 is the decreasing complete right series which begins with px(3p-2).

Also

199 193 181 163 139 109 73 31 107 101 89 71 47 17

79 73 61 43 19

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are special incomplete decreasing series

131 137 149 167 191 is an incomplete increasing series

2) For left primes there are no complete series because at each step the prime is added to the base below , the series gives at some step a composite. e.g primes ending with 7 to the right give composite in the second step. Ending with 9 to the right give composites in the first step. Ending with 1 give composites at different steps,31 at second step,61 at 4th step. . . Also ending with 3 give composites at different steps, 13 at third step, 43 ending at first step,73 ending at second step,103 at 7th step, 113 ending at 5th step. . . ..

With similar arguments there are no decreasing complete series in this case 3) The base in this case is P

6k. k= 1 till k=1,2,3,. . . ..

Base: 6 18 36 60 90 126 168 216 270 330 396 468

546 630 720 816 918 1026 1140. . . .

4) There are infinite number of incomplete right or left series similar to the case with b=

2.

Examples:

5 41 83 131

11 59 113 173 239 311 389

23 53 89 131 179 233 293

7 31 61 97 139

19 43 73 109 151 199

37 67 109 157 211 271 337 409 487 571 661 757 859

967 1081 1201 1327 1459 1597 1741 E Application Example

1- The arithmetic series which give the composites till 130 are four given by:

x=n2 + 2nk where n= 3, 5, 7, 11

The linear arithmetic series which give the primes in the same range are four given by

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p +P

2k where k =1 till 1. . ...p-2 and p = 11,17, 41 and 109 107 103 97 89 79 71 67 53 37 19

2- In case of right primes till 130 we have four series:

p +P

6k , k=0 till k=0. . . p-2 p=11, 23 107 -P

6k , k= 0 till p=0. . . 5 5 +P

6k , k=0 till k=0, 8. . .11

3- In case of left primes we have also four series:

p +P

6k , k=0 till k= 0,4. . .6 p=7,13, 19 37 +P

6k , k=0 till k= 0,7. . .23

F Remarks

1) For high value of x there are no complete linear arithmetic series so that more incom- plete series are used to get the primes in a certain range. The incomplete series can be at least of 3 primes.

2) The result can be formulated as follows: Every prime≥ 5 is a member of one or more linear arithmetic series with number of primes ≥ 3.

3- CONCLUSIONS

It was shown that primes ≥ 5 have the structure of linear arithmetic series with step increasing by 2. Also if the concept of left and right

Primes is used the step increase is 6. Complete and incomplete series are used. It was shown that this is valid for a small range of primes

≥ 5. For very large range we expect the same result with relatively more incomplete series. This is not proven in general here.

REFERENCES

1) H. Riesel. Prime numbers and computer methods for factorization. Birkhauser, Basel, 1. Edition, 1994.

2) M. Mansour. Special semigroups in elementary number theory. International journal of pure and applied mathematics. Vol.57 No. 3,

pp. 419-433, 2009.

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3) M. Mansour. New characterization of primes and its consequences. 2017, URL : https://www.research-collection.ethz.ch/handle/20.500.11850/178874 .

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