Predicting the Primes 1.

A Reasoned Exposition of the Distribution of the Composite Numbers

Abstract

A large part of the distribution of the composite numbers can be identified simply by inspection. All numbers ending in an even number or 5 can be simply read off the number series as composite numbers. This is because they are products of 2 or 5, both primes which are factors of 10, which is the cyclical basis of our number system. The distribution of the remaining composite numbers is not simply readable, but is equally predictable because it is created by similar cycles of multiplication on each of the primes that is not a factor of 10 (that is, other than 2 and 5). The distribution of the primes is, of course, determined by the distribution of the composite numbers.

 

Our number system is, entirely fortuitously, built on repeated decades. This exercises quite specific, but also fortuitous, constraints on the patency of divisibility of numbers, that is, on their character as composite or prime numbers.

Half of the numbers of each decade are divisible by 2. These are the even numbers and represent half of all numbers. The only other number within the first decade that divides evenly into it is 5, creating, like 2, a predictable and recognizable pattern of composite numbers throughout the number system. Thus:

The number 2 makes every second number, that is, the even numbers, a composite number.

The number 5 makes every fifth number, that is, ending in 5 or 0, a composite number, although the 0-ending numbers have already been accounted for by 2.

Because the primes 2 and 5 are factors of 10, which is the repeating base of our number system, we can recognize the composite numbers of which they are factors by inspection – numbers ending in 2, 4, 5, 6, 8, or 0. Thus the majority of the composite numbers are recognizable by inspection, and we also know that all other primes can only end in 1, 3, 7, or 9. All numbers ending in these digits are either primes or the products of primes other than 2 and 5.

Like 2 and 5, each other number in the “total series” will determine a regular pattern of composite numbers as it multiplies first itself, and then each number above it. Numbers that are built on the two primes that are factors of 10 (e.g. 4, 25) continue to be immediately detectable on the basis of their final digit – an even number or 5. Those based on a prime which is not a factor of 10 cannot be so simply identified. They will end in 1, 3, 7, or 9, but numbers with these as final digits may be either primes or composite numbers. What is it, as an innate characteristic of out number system, that determines their individual status as prime or composite numbers?

The remaining primes of the first decade – 3 and 7 – although not factors of 10, nevertheless also subtend recurring patterns of composite numbers, which extend beyond the first decade, and which, cycling in a manner determined by the decade structure of the number system, also, eventually, return to scratch at the start of a decade. The composite numbers built on each – its products, whether primes, composite numbers or a combination of the two – will form, and be recognizable in, a cyclical pattern determined by the pace (length of number series) at which one of its products first, and regularly thereafter, coincides with a root of the decade: that is, when a product it creates ends in 0, and this will only occur when 10x is reached. Thus its pattern will only begin to repeat beyond 10x, and thereafter at 20x, 30x, 40x … and so on.

This is initially, and most easily, seen in the repeating pattern of composite numbers built on the prime 3, in a cycle 30 (i.e. 1 – 30, 31 – 60, 61 – 90 etc), but similarly occurs, forming a different pattern, for those deriving from the prime 7, which cycle between 1 and 70 (71 – 140, 141 – 210 etc). 10x is always the first point at which a number, and for our purposes a prime number (e.g. 3, 7, 11), reassumes the initial position in a decade, and from that point the pattern of its creation of composite numbers will recur until the next sequence is completed. This is an essential concept, permitting us to see the products of each prime number as recurring patterns within our number system based on 10.

Applying the above: the prime number 3 does not divide evenly into 10, but it, like 2 and 5, determines a regular pattern of divisibility in the numbers which follow it, and its regular cycle extends over not one, but three decades; that is, the cycle is from 1 to 30. Unlike those deriving from 2 and 5, however, the composite numbers built on it are not identifiable by simple inspection. The number series 1 – 89 (three cycles), showing only numbers ending in 1, 3, 7, and 9 (since all others end in even numbers or 5 and so are already identified as composite), is set out below and the pattern of composite numbers built on 3, that is, when 3 is multiplied first by itself and then by each following number which is neither even nor 5 (which products can be simply read as composites), is shown in bold:

1          3          7          9

11        13        17        19

21        23        27        29

 

31        33        37        39

41        43        47        49

51        53        57        59

 

61        63        67        69

71        73        77        79

81        83        87        89

It can be seen that the pattern repeats over each successive set of 30 numbers. It will recur ad infinitum, as 3 is successively multiplied by each number ending in 1, 3, 7, and 9. It is immediately recognizable that the potential primes remaining after removing even numbers and numbers ending in 5, are, after the identification of the composite numbers based on 3, reduced by a further 4 per 3 decades; that is, from 12 (all numbers ending in 1, 3, 7, 9) to 8. In every successive 30 numbers, a maximum of 8 can now be prime, due to the multiplicative outcomes of the prime numbers 2, 3, and 5.

Similarly, for the prime number 7 we have the following repeating pattern over cycles of 70 numbers:

1          3          7           9         11        13        17        19        21        23         27       29

31        33        37        39        41        43        47        49        51        53        57        59

61        63        67    69

 

71        73        77        79        81        83        87        89        91        93        97        99

101      103      107      109      111      113      117      119      121      123      127      129

131      133      137      139

(The numbers in bold italics are those already identified as composite numbers for the cycle of 3.)

It will be noticed that all the numbers ending in 1, 3, 7, and 9 not identified as composite numbers in the recurring cycles of 3 and 7 are primes (with the exception of 121, which will be identified in the next cycle, based on 11). Each of these primes in turn will create its own cyclic pattern of composite numbers, so determining the complete (within any set of limits) distribution of composite numbers; and as above, those numbers left out of the distribution form the next in the series of primes, and will be the bases of further cycles of the non-readable composite numbers.

That is, every succeeding prime number creates, by its multiplicative action, a regular pattern of composite numbers within its cycle of 1 to 10x, and will, reliably according to its pattern, create/identify composite numbers among the numbers left as potential primes after the extraction of the products of all lower primes

Moving up the primes, beyond the first decade, a repeating cycle will occur for 11 over 11 decades (1 – 110, and thereafter), and for 13 over 13 decades (1 – 130 and thereafter).

Thus for the prime number 11 we have the following repeated pattern in a cycle of 110:

1          3          7           9         11        13        17        19        21        23         27       29

31        33        37        39        41        43        47        49        51        53        57        59

61        63        67        69        71        73        77        79        81        83        87        89

91        93        97        99        101      103      107      109

 

111      113      117      119      121      123      127      129      131      133     137      139

141      143      147      149      151      153      157      159      161      163      167      169

171      173      177      179      181      183      187      189      191      193      197      199

201      203      207      209      211      213      217      219

 

(Again, the numbers in bold italics have already occurred as composite numbers in the cycles of  the two lower primes, 3 and 7.)

It will be observed that in each cycle, whether of 30, 70 or 110, four numbers are taken out as composite numbers by the given base number, deriving from the four numbers ending in 1, 3, 7, and 9 by which the prime is multiplied before the end of the cycle is reached.

Each successive prime number creates a regular but different pattern of composite numbers as this process proceeds up the scale, with each cycle and pattern repeating itself ad infinitum. As the base number increases in size, so does its cycle size, but each of these cycles, regardless of its length, introduces four non-readable composite numbers (see tables above), although varying numbers of these are novel (that is, not already created/identified as composite numbers, by a lower prime).

If we superimpose all such cycles, with their differing patterns of composite numbers, on one another (having already eliminated all even numbers and numbers ending in 5), we will have as remainder the distribution of primes in the total number series. That is, those numbers in the total series that do not occur as products (that is, as composite numbers) in this successive process are the prime numbers. The distribution of the primes is thus determined by the distribution of the composite numbers.

The following table shows this process of superimposition for the cycles of 3, 7, and 11 to the end of the second cycle of 11 (the composite numbers deriving from 2 and 5 are again omitted as entirely immediately recognizable), The numbers shown in bold are those which occur as products in the cycles of 3, 7, and 11; that is, they are all composite numbers:

 

1          3          7           9         11        13        17        19        21        23         27       29

31        33        37        39        41        43        47        49        51        53        57        59

61        63        67        69        71        73        77        79        81        83        87        89

91        93        97        99        101      103      107      109      111      113      117      119

121      123      127      129      131      133     137       139      141      143      147      149

151      153      157      159      161      163      167      169*    171      173      177      179

181      183      187      189      191      193      197      199      201      203      207      209

211      213      217      219

*the first non-redundant appearance of 13

 

It can be seen that all the remaining numbers, after the composite numbers identified by the five number cycles (2, 3, 5, 7, 11) are eliminated, are prime numbers. (The first of the autonomous cycle of 13 occurs within this range and is also identified as a composite number).

 

The cyclic patterns of composite numbers built on primes, in the range 1 to 2000

The full complement of composite numbers in fact represents the successive multiplication of each number >1 in the “total series” by each other number in turn, thus: 2 x 2, 2 x 3, 2 x 4 etc., then 3 x 2, 3 x 3, 3 x 4 etc., then 4 x 2, 4 x 3, 4 x 4 etc., and so on. These calculations are, however, redundant for any number lower than the multiplier, as that lower number will already have contributed the same composite when it took its turn as multiplier. Thus the autonomous 5 series begins at 25, as 2, 3, and 4 as multipliers have already disposed of 10, 15, and 20.  5 then takes on this role for higher number multipliers – 6, 7 etc. Therefore, in what follows, the table begins with the square of the number concerned. Further to avoid redundancy, as already explained, only primes are included as base numbers, and only numbers ending in 1, 3, 7, and 9 are used as multipliers.

As a larger demonstration, the following tables, and their later superimposition, cover the number series from 0 to 2000, and show the composite numbers derived from all primes up to 43; the next prime, 47, squared is 2209, and so falls outside the range 0 – 2000. Composite numbers derived from 2 and 5, identifiable by simple inspection, are not shown, and composite numbers created by 3 as well as by the table multiplier are in italics.

It can be seen that, in every case, each row (showing the 4 composite numbers created in each cycle) exhibits the same recurring pattern, which can be simply written down, ascending the cycles, without the need for calculation, just as one can write down the composite numbers based on 2 – 2,4,6,8,10 / 12,14,16,18,20 – ascending the number series, without calculation.

 

Composite numbers deriving from 7

7 – 2000 :  29 cycles of 70                                                                               7 squared = 49

successive multipliers ending in

1                      3                      7                      9

0-69                                                                             49                    63

70-139                         77 (7×11)         91(7×13)          119(7×19)        133(7×19)

140-209                       147(7×21)        161(7×23)        189(7×27)        203(7×29)

210 etc                         217 (etc)          231                  259                  273

280                              287                  301                  329                  343

350                              357                  371                  399                  413

420                              427                  441                  469                  483

490                              497                  511                  539                  553

560                              567                  581                  609                  623

630                              637                  651                  679                  693

700                              707                  721                  749                  763

770                              777                  791                  819                  833

840                              847                  861                  889                  903

910                              917                  931                  959                  973

980                              987                  1001                1029                1043

1050                            1057                1071                1099                1113

1120                            1127                1141                1169                1183

1190                            1197                1211                1239                1253

1260                            1267                1281                1309                1323

1330                            1337                1351                1379                1393

1400                            1407                1421                1449                1463

1470                            1477                1491                1519                1533

1540                            1547                1561                1589                1603

1610                            1617                1631                1659                1673

1680                            1687                1701                1729                1743

1750                            1757                1771                1799                1813

1820                            1827                1841                1869                1883

1890                            1897                1911                1939                1953

1960                            1967                1981

 

Composite numbers deriving from 11

11 – 2000 :  19 cycles of 110                                                                           11 squared = 121

successive multipliers ending in

1                      3                      7                      9

0

110                              121                  143                  187                  209

220                              231                  253                  297                  319

330                              341                  363                  407                  429

440                              451                  473                  517                  539

550                              561                  583                  627                  649

660                              671                  693                  737                  759

770                              781                  803                  847*                869

880                              891                  913                  957                  979

990                              1001                1023                1067                1089

1100                            1111                1133                1177                1199

1210                            1221                1243                1287                1309*

1320                            1331                1353                1397                1419

1430                            1441                1463*              1507                1529

1540                            1551                1573                1617                1639

1650                            1661                1683                1727                1749

1760                            1771*              1793                1837                1859

1870                            1881                1903                1947                1969

1980                            1991

 

*composite numbers based on 7 also in italics

 

Composite numbers deriving from 13

13 – 2000 :  16 cycles of 130                                                                           13 squared = 169

successive multipliers ending in

1                      3                      7                      9

0

130                                                      169                  221                  247

260                              273                  299                  351                  377

390                              403                  429                  481                  507

520                              533                  559                  611                  637*

650                              633                  689                  741                  767

780                              793                  819                  871                  897

910                              923                  949                  1001*              1027

1040                            1053                1079                1131                1157

1170                            1183*              1209                1261                1287

1300                            1313                1339                1391                1417

1430                            1443                1469                1521                1547*

1560                            1573#              1599                1651                1677

1690                            1703                1729*              1781                1807

1820                            1833                1859                1911                1937

1950                            1963                1989

 

#composite numbers based on 11 also in italics

 

Composite numbers deriving from 17

17 – 2000 :  12 cycles of 170                                                                           17 squared = 289

successive multipliers ending in

1                      3                      7                      9

0

170                                                                              289                  323

340                              357                  391                  457                  493

570                              527                  561                  629                  663

680                              697                  731                  799                  833*

850                              867                  901                  969                  1003

1020                            1037                1071                1139                1173

1190                            1207                1241                1309*              1343

1360                            1377                1411                1479                1513

1530                            1547*              1581                1649                1683

1700                            1717                1751                1819                1853

1870                            1887                1921                1989

 

Composite numbers deriving from 19

19 – 2000 :  11 cycles of 190                                                                           19 squared = 361

successive multipliers ending in

1                      3                      7                      9

0

190                                                                                                      361

380                              399                  437                  513                  551

570                              589                  627                  703                  741

760                              779                  817                  893                  931*

950                              969                  1007                1083                1121

1140                            1159                1197                1273                1311

1330                            1349                1387                1463*              1501

1520                            1539                1577                1653                1691

1710                            1729*              1767                1843                1881

1900                            1919                1957

 

Composite numbers deriving from 23

23 – 2000 :  9 cycles of 230                                                                             23 squared = 529

successive multipliers ending in

1                      3                      7                      9

0

230

460                                                      529                  621                  667

690                              713                  759                  851                  897

920                              943                  989                  1081                1127*

1150                            1173                1219                1311                1357

1380                            1403                1449                1541                1587

1610                            1633                1679                1771*              1817

1840                            1863                1909                                       

 

Composite numbers deriving from 29

29 – 2000 :  7 cycles of 290                                                                             29 squared = 841

successive multipliers ending in

1                      3                      7                      9

0

290

580                                                                                                      841

870                              899                  957                  1073                1131

1160                            1189                1247                1363                1421*

1450                            1479                1537                1653                1711

1740                            1769                1827                1943

Composite numbers deriving from 31

31 – 2000 :  7 cycles of 310                                                                             31 squared = 961

successive multipliers ending in

1                      3                      7                      9

0

310

620

930                              961                  1023                1147                1209

1240                            1271                1333                1457                1519*

1550                            1581                1643                1767                1829

1860                            1891                1953

 

Composite numbers deriving from 37

37 – 2000 :  6 cycles of 370                                                                             37 squared = 1369

successive multipliers ending in

1                      3                      7                      9

0

370

740

1110                                                                            1369                1443

1480                            1517                1591                1739                1813*

1850                            1887                1961

Composite numbers deriving from 41

41 – 2000 :  5 cycles of 410                                                                             41 squared = 1681

successive multipliers ending in

1                      3                      7                      9

0

410

820

1230

1640                            1681                1763                1927

Composite numbers deriving from 43

43 – 2000 :  5 cycles of 430                                                                             43 squared = 1849

successive multipliers ending in

1                      3                      7                      9

0

430

860

1290

1720                                                    1849

 

47 squared goes beyond the matrix.

 

Superimposition of distributions of composite numbers based on primes 3 and 7 – 43

When all 11 series are superimposed we get the matrix of composite numbers shown in the following table in which all composite numbers are shown in bold.  It can be seen that the remaining (non-bold) numbers are the primes.

Distribution of Composite Numbers (in bold) and Primes in the Number Series 1 to 2000 (not including composite numbers based on 2 and 5)

3                      7                     9

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

100                  01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

200                  01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

300                  01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

400                  01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

500                  01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

600                  01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

700                  01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

800                  01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

900                  01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

1000                01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

1100                01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

1200                01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

1300                01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

1400                01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

1500                01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

1600                01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

1700                01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

1800                01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99

1900                01                    03                    07                    09

11                    13                    17                    19

21                    23                    27                    29

31                    33                    37                    39

41                    43                    47                    49

51                    53                    57                    59

61                    63                    67                    69

71                    73                    77                    79

81                    83                    87                    89

91                    93                    97                    99                    2000

           

Distribution of the primes

It can be seen that the distribution of the composite numbers, as expounded above, determines the distribution of the primes.

The increasing entry of new cycles into the system means that the incidence of primes decreases as the number series advances. However, the increasing length of cycle for each successive number matrix that enters the system means that the rate at which the incidence of primes diminishes decelerates as the series advances. An additional factor is that the level of redundancy of the four new composite numbers per cycle must increase as more cycles enter the system.

The fact that 5 of the 8 potential primes per 30 numbers remaining after the extraction of the multiples of 2, 3, and 5 are consecutive predicts the clustering of primes that occurs from time to time, despite a diminishing incidence.

If there were to be a highest prime number, it would occur when sufficient cycles had entered the system to entirely choke out the residual of the potential 8 primes per cycle of 30 when the cycles of 2, 3 and 5 have been accounted for. Whether this can occur (and proofs suggest that it does not) depends on the rate at which composite numbers are generated as against the rate at which new numbers enter the system.

Using this approach, the primes do not have to be identified post hoc, as it were, but will automatically emerge from increasing the extension of the matrix of composite numbers which is created by pursuing the cycling of the products of the successive primes.

 

November  2003

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