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

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

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

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

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

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

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

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

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

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

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

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

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

Distribution of the primes

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