_{1}

^{*}

The prime numbers P≥5 obey a pattern that can be described by two forms or geometric progressions or that facilitates obtaining them sequentially, being possible also to calculate the quantity of primes that are in the geometric progressions as it is described in this document.

Since ancient times when humans discovered the counting system and natural numbers, prime numbers immediately attracted their attention, were numbers whose only divisors are 1 and the same number. The problem to find them was not to be able to describe by means of an equation. There are countless publications about the properties of prime numbers that can be found in all languages and theorems have been created in different ways, seeking always to find a pattern of ordering [

“The succession of primes is unpredictable. We don’t know if they will obey any rule or order that we have not been able to discover still. For centuries, the most illustrious minds tried to put an end to this situation, but without success. Leonhard Euler commented on one occasion: mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers and we have reason to believe that it is a mystery into which the human mind will never penetrate. In a lecture given by D. Zagier in 1975, he said: “There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, [they are] the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision.” (Havil, 2003 [

To put prime numbers into context, let’s begin by saying anecdotally, as late as 20,000 years ago humans marked the bone of Ishango with 19, 17, 13, 11 [

Until now, there is no known efficient formula for primes, nor a recognizable pattern or sequence the primes follow. All recent publications dealing with this issue established that primes are distributed at random and looked more to a white noise distribution [

Porras Ferreira and Andrade (2014) [

Or all prime numbers

It can be seen that Equation (2) is a derivation of Equation (1), as demonstrated in [

Equation (2) can be transformed to a simpler form where

It means, the form of all prime numbers only have these three equations: Equation (1) which does not include primes 2, 3 and 5, and Equation (2) and Equation (3) which does not include primes 2 and 3, therefore Equation (2) and Equation (3) are equivalent:

Taking the Equation (3) a table is constructed (

In the following analysis,

Theorem 1:

If

Demonstration:

Let

Factoring

which must be a composite number with two factors

The theorem is proved.

Corollary 1:

The composite numbers

Corollary 2:

There are not identical composite numbers, one from column

Corollary 3:

All the composite numbers

Corollary 4:

All the composite numbers N_{k} of column 6k + 1 may have two factor from the column 6k − 1, that means

Corollary 5:

Eliminating all k, product of Equation (5), the rest k will only contain primes of the given form of Equation (3) and as it is shown in

1) The prime

2) The prime

3) The prime

4) The prime

5) From the analysis of the previous 4 points and applying Theorem 1 we can conclude the following with respect to Equation (5) being the geometric progression

With

They contain composite numbers of the form

And

They contain prime numbers of the form

In Equation (6) there exist symmetry of cells where

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |

1 | 6 | 13 | 20 | 27 | 34 | 41 | 48 | 55 | 62 | 69 | 76 | 83 | 90 | 97 | 104 |

2 | 11 | 24 | 37 | 50 | 63 | 76 | 89 | 102 | 115 | 128 | 141 | 154 | 167 | 180 | 193 |

3 | 16 | 35 | 54 | 73 | 92 | 111 | 130 | 149 | 168 | 187 | 206 | 225 | 244 | 263 | 282 |

4 | 21 | 46 | 71 | 96 | 121 | 146 | 171 | 196 | 221 | 246 | 271 | 296 | 321 | 346 | 371 |

5 | 26 | 57 | 88 | 119 | 150 | 181 | 212 | 243 | 274 | 305 | 336 | 367 | 398 | 429 | 460 |

6 | 31 | 68 | 105 | 142 | 179 | 216 | 253 | 290 | 327 | 364 | 401 | 438 | 475 | 512 | 549 |

7 | 36 | 79 | 122 | 165 | 208 | 251 | 294 | 337 | 380 | 423 | 466 | 509 | 552 | 595 | 638 |

8 | 41 | 90 | 139 | 188 | 237 | 286 | 335 | 384 | 433 | 482 | 531 | 580 | 629 | 678 | 727 |

9 | 46 | 101 | 156 | 211 | 266 | 321 | 376 | 431 | 486 | 541 | 596 | 651 | 706 | 761 | 816 |

10 | 51 | 112 | 173 | 234 | 295 | 356 | 417 | 478 | 539 | 600 | 661 | 722 | 783 | 844 | 905 |

11 | 56 | 123 | 190 | 257 | 324 | 391 | 458 | 525 | 592 | 659 | 726 | 793 | 860 | 927 | 994 |

12 | 61 | 134 | 207 | 280 | 353 | 426 | 499 | 572 | 645 | 718 | 791 | 864 | 937 | 1010 | 1083 |

13 | 66 | 145 | 224 | 303 | 382 | 461 | 540 | 619 | 698 | 777 | 856 | 935 | 1014 | 1093 | 1172 |

14 | 71 | 156 | 241 | 326 | 411 | 496 | 581 | 666 | 751 | 836 | 921 | 1006 | 1091 | 1176 | 1261 |

15 | 76 | 167 | 258 | 349 | 440 | 531 | 622 | 713 | 804 | 895 | 986 | 1077 | 1168 | 1259 | 1350 |

In

and

The number of times the above occurs in_{1} ≠ m_{2}

This is important for calculating the number of primes smaller or equal to N_{k}, as will be seen later.

1) The prime

1 | 2 | 3 | 4 | 5 | 6 | 7 | |||

1 | 6 | 13 | 20 | 27 | 34 | 41 | 48 | 6, 11, 13 | 1, 2, 3, 4, 5 |

2 | 11 | 24 | 37 | 50 | 16, 20, 21 | 7, 8, 9, 10, 12 | |||

3 | 16 | 35 | 24, 26, 27 | 14, 15, 17, 18 | |||||

4 | 21 | 46 | 31, 34, 35 | 19, 22, 23, 25 | |||||

5 | 26 | 36, 37, 41 | 28, 29, 30, 32 | ||||||

6 | 31 | 46, 48, 50 | 33, 38, 39, 40 | ||||||

7 | 36 | 51 | 42, 43, 44, 45 | ||||||

8 | 41 | 47, 49 | |||||||

9 | 46 | ||||||||

10 | 51 |

2) The prime

3) The prime

4) The prime p = 11 from column 6k − 1 appears for the first time in the row

5) From the analysis of the previous 4 points and applying Theorem 1 we can conclude the following with respect to Equation (5) being the geometric progression

With

With

where

Therefore:

They contain composite numbers of the form

They contain prime numbers of the form

In

Also in

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

1 | 8 | 15 | 22 | 29 | 36 | 43 | 50 | 57 | 64 | 71 |

2 | 15 | 28 | 41 | 54 | 67 | 80 | 93 | 106 | 119 | 132 |

3 | 22 | 41 | 60 | 79 | 98 | 117 | 136 | 155 | 174 | 193 |

4 | 29 | 54 | 79 | 104 | 129 | 154 | 179 | 204 | 229 | 254 |

5 | 36 | 67 | 98 | 129 | 160 | 191 | 222 | 253 | 284 | 315 |

6 | 43 | 80 | 117 | 154 | 191 | 228 | 265 | 302 | 339 | 376 |

7 | 50 | 93 | 136 | 179 | 222 | 265 | 308 | 351 | 394 | 437 |

8 | 57 | 106 | 155 | 204 | 253 | 302 | 351 | 400 | 449 | 498 |

9 | 64 | 119 | 174 | 229 | 284 | 339 | 394 | 449 | 504 | 559 |

10 | 71 | 132 | 193 | 254 | 315 | 376 | 437 | 498 | 559 | 620 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

1 | 4 | 9 | 14 | 19 | 24 | 29 | 34 | 39 | 44 | 49 |

2 | 9 | 20 | 31 | 42 | 53 | 64 | 75 | 86 | 97 | 108 |

3 | 14 | 31 | 48 | 65 | 82 | 99 | 116 | 133 | 150 | 167 |

4 | 19 | 42 | 65 | 88 | 111 | 134 | 157 | 180 | 203 | 226 |

5 | 24 | 53 | 82 | 111 | 140 | 169 | 198 | 227 | 256 | 285 |

6 | 29 | 64 | 99 | 134 | 169 | 204 | 239 | 274 | 309 | 344 |

7 | 34 | 75 | 116 | 157 | 198 | 239 | 280 | 321 | 362 | 403 |

8 | 39 | 86 | 133 | 180 | 227 | 274 | 321 | 368 | 415 | 462 |

9 | 44 | 97 | 150 | 203 | 256 | 309 | 362 | 415 | 468 | 521 |

10 | 49 | 108 | 167 | 226 | 285 | 344 | 403 | 462 | 521 | 580 |

The number of cells repeated in each column

The cell symmetry in Equation (8) can be eliminated by taking only the values of

They contain composite numbers of the form

They contain prime numbers of the form

In some cases

Regardless of how to find primes with Equation (7) and Equation (11), there is the traditional test for whether or not a number is prime, using Theorem 2.

1) For

Theorem 2:

If

Demonstration:

Let N_{k} > 5 be a composite integer of the form

1 | 2 | 3 | ||||||

1 | 8 | 4 | 4, 8, 9 | 1, 2, 3, 5, 6 | ||||

2 | 15 | 28 | 9 | 20 | 14, 15, 19 | 7, 10, 11, 12 | ||

3 | 22 | 41 | 14 | 31 | 48 | 20, 22, 24 | 13, 16, 17, 18 | |

4 | 29 | 19 | 42 | 28, 29, 31 | 21, 23, 25, 26 | |||

5 | 36 | 24 | 34, 36, 39 | 27, 30, 32, 33 | ||||

6 | 43 | 29 | 41, 42, 43 | 35, 37, 38, 40 | ||||

7 | 50 | 34 | 44, 48, 49 | 45, 46, 47, 51 | ||||

8 | 39 | 50 | ||||||

9 | 44 | |||||||

10 | 49 |

As

As it is known that for

Example: primes (11, 17, 23, 29, and 41). The problem with Equation (12) is that its application is more difficult for very large primes; one would have to know all primes smaller than

2) For the

Applying the same theorem shown for the

Example: primes (13, 19, 31, and 37). Similarly, the application of Equation (13), is more difficult for very large primes, while the application of Equation (10) and Equation (11), is simpler to obtain all

1) The total of primes

2) The number of primes _{1} ≠ m_{2},

The term

Example of

then

Example of

and

therefore

3) The number of primes

The term

its factors, the term

peated cells in

to the number of repeated cells where _{m}_{1}p_{m}_{2}p_{n} with

Example of

With

Example of

With

4) With

All prime numbers

Verification of whether a number is prime can be done in

The primes do not appear in random form, their sequence is determined by the Equation (6) and Equation (7) in

The number of primes

To all my professors from Escuela Naval de Cadetes Colombia and the Naval Postgraduate School of the United States (Naval Postgraduate School, Monterey California USA).

Porras Ferreira, J.W. (2017) The Pattern of Prime Numbers. Applied Mathematics, 8, 180-192. https://doi.org/10.4236/am.2017.82015