Prime Numbers Without Mystery 2

1 PRIME NUMBERS WITHOUT MYSTERY 2

Jânia Duha Universidade Federal do Paraná – UFPR Curitiba, Brasil

ABSTRACT Prime numbers have been quoted since ancient times as a mystery. Through the centuries much has been accomplished and there are several interesting works on this theme. One of the most interesting things about primes is that despite the complexity of some models one should not forget that prime numbers are natural numbers. Seeking the lost simplicity of ancient prime numbers theories, we will follow a slight different path, and yet, very similar to the one presented by Erathostenes, more than 2000 years ago. The model presented here was first developed at 2002. At that time, it became astonishing that words like mystery were still in use concerning the prime numbers literature. As a result in 2004 the first Prime numbers without mystery was released (Duha, 2004, 2004b) and now the updated version is being presented here. There is more information concerning the periods of the knot numbers and a closer view to the prime numbers patterns. The periodicity of these patterns yields as the key to the understanding of prime numbers. This work offers an explanation to why Prime-numbers look randomly at first sight, or else, why the pattern cannot be easily seen although it is there all the time. Finely, we conclude that: prime numbers do obey laws, very simple, indeed, do follow patterns, very predictable, and can be completely explained, understood and predicted.

THE LOGICAL PATH - MODELING The idea behind the Knot Model (Duha, 2002, no publication) is very simple. However as every one that works with science knows, “do it simple” is not so easy at all. Every time you don’t understand something your first solution is, in general, very complicated. The most simple is a model, the better. In fact, the logical path presented here is the result of several hours of work and brain storms, because, at the beginning it was not so simple, but after taking a second, third, fourth, etc. look to the initial ideas, new aspects rose up from the darkness of the unknown and at the end, it became as simple as a first order equation can be. The logical path showed itself very similar to the one that produced the Sieve of Eratosthenes, but with some fundamental differences. These differences will allow us to answer such questions as: (a) where, prime numbers can be found among the natural sequence of numbers? (b) how one can be 100% sure that the output is a prime number without any test of primality? (c) are there patterns that rules its behavior? …and hundred of others questions not so important, but still interesting, like: what are twin primes? Now, let’s start from the beginning. First of all: a prime number is a natural number. Natural numbers are a very simple sequence of all possible numbers that you will find if you add 1 to the previous number in the sequence: 1, 2, 3, 4 …. The number one is a construct number. By adding one successively you construct all the sequence. But the numbers two and three are still “small” enough to be used as constructing blocks and they can provide a good part of the natural numbers sequence as follows

1

2 2

3 3

4 4

5

6 6 6

7

8 8

9 9

10 10

11

12 12 12

13

14 14

15 15

16 16

17

18 18 18

19

20 20

21 21

22 22

23

24 24 24

25

26 26

27 27

28 28

29

30 30 30

... 31

Now, let us point to the very simple fact that there is a part of the natural sequence that cannot be provided by the 2 and the 3. When one tries to reproduce the natural sequence of numbers using the 2 and the 3 as construct numbers, the first thing to be noted is that there is something missing! And, on the other hand, at the same time, there is an overload of information at some points. The two and the three provides simultaneously the numbers 6, 12, 18, etc…We will call these repeated numbers as the "knots" of the sequence (in green) as follows

1

2 2

3 3

4 4

5

6 6 6

7

8 8

9 9

10 10

11

12 12 12

13

14 14

15 15

16 16

17

18 18 18

19

20 20

21 21

22 22

23

24 24 24

25

26 26

27 27

28 28

29

30 30 30

... 31

2 Note also, that these knots have always an odd number on the left and on the right side that cannot be provided by the two or the three (in red). Let us call this special class of odd numbers around the knots as knotnumbers (kn). If you look more carefully, you will note that some of these numbers are simple (can be divided only by 1 and by their selves) and some are composite, or else, they are multiples of other knot-numbers (but only! knotnumbers). So, the numbers around the knots can be simple (ks) or composite (kc). Note that, the ks numbers are also known as prime numbers. The knots, the knot-numbers and the composite knot-numbers are given by the following equations, respectively (1) k = 6ni , (2) kn = 6ni ± 1, (3) kc = (6ni ± 1) (6nj ± 1), with ni = 1,2,3,4,5…, nj = 1,2,3,4,5… The simple knot-numbers (prime numbers) are “always” knot-numbers but “never” composite knotnumbers, or else, the sequence of prime numbers is given by (4)

Π ks = Π kn - Π kc

To find the knots with their respective knot-numbers don not requires any computational work. As a consequence, the mapping of the primes (ks numbers) relies completely on the mapping of the kc numbers. So let’s pay more attention to the kc numbers and its periodicities. The kc numbers are composed by the multiplication of knot-numbers solely, or else: 5x5=25, 5x7=35, 5x11=55, 5x13=65, 5x17=85, etc.; and 7x7=49, 7x11=77, 7x13=91, etc.; and 11x11=121, 11x13=143, etc.; and so on, for all the sequence of knot-numbers, as shown at Table 1.

Table 1 – First contributions of the kc numbers.

5

7

25

5

11 55

35

11 13 17 19

17

19

85 65

49

7

13

23 115

95

91

25

29 145

175

42

77

119

161

203

121

187

253

319

143

209

275

169

247

325 299

377

391

493 425 475

437

66 78

289

361

28-14

22-44

221

323

30 10-20

125

133

Periodicity

52-26

102 34-68

114 551

76-38

The kc numbers can be easily calculated (Eq. 3) and it is also, very easy to predict their behavior through the infinity because they present a clear periodicity. The sequence of knot-numbers with their respective knots is even easier to obtain and to predict. The knots start at 6 and will appear always at intervals of six also: 6, 6+6=12, 12+6=18, 18+6=24, 24+6=30, etc. Note, that you can be 100% sure that all known and unknown prime numbers will be located around a knot. A prime number is a knot-number! Twin primes are knot-numbers that share the same knot! All that you have to do is to discard the composite knot-numbers. After that, all numbers remaining around the knots will be prime numbers with 100% of certainty and no need of any test. This is very similar to the Eratosthenes sieve, however, note that there is a fundamental difference here. With the Eratosthenes sieve you cannot switch numbers when calculating primes, but you can with the knot model. The Eratosthenes sieve works well for small numbers but it became too heavy and useless when going to big numbers. The knot model hasn’t this limitation and more, as we will try to show as follows.

3 PRIME NUMBERS PATTERN

Figure 1 shows five bar-code patterns. The first pattern from the bottom is the kn pattern which has a period Tn = 6, or else, starting at 6 (first knot), every point at 6 + 6n (with n=1,2…∞) is also a knot that, contributes with two knot-numbers for the pattern. The second pattern is the kc(5) pattern with all multiples of 5 for the interval we are focusing (5 to 469). The kc(5) period is T5 = 30 with two contributions at every 52 + 30n. The third pattern is the kn(-5), or else, the kn pattern after subtracting the kc(5) contributions. The combination of these two patterns will result on a new pattern for the kn numbers with Tn = 30. The fourth pattern is the kc(7) contribution with T7 = 42 and the fifth pattern is the kn pattern after subtracting the kc(7) with Tn = 210. FIGURE 1 - Bar-code pattern for kn & kc numbers

kn(-5,-7)

kc(7)

kn(-5)

kc(5)

kn

5

25

49

259

469

Figure 1. From the bottom to the top: (a) kn numbers (two for every knot); (b) kc(5) numbers are the multiples of 5 starting at 52 = 25; (c) kn(-5) numbers after subtracting the kc(5) contributions; (d) kc(7) numbers are the multiples of 7 starting at 72 = 49; (e) kn(-5,-7) numbers after subtracting the kc(5) and kc(7) contributions.

Every time you insert the contribution of a new set of multiples of a knot-number, the resulting pattern changes and the new period quickly increases: Tn ( −5) = 6 × 5 = 30 Tn ( −5,−7) = Tn ( −5) × 7 = 210 Tn ( −5,−7,−11) = Tn ( −5,−7) × 11 = 2310 , etc.. so that, every knot-number you consider, to built the composite knot-numbers and clean up the kn pattern will make the period of the resulting pattern increase at approximately one order of magnitude. The fifth pattern at Figure 1 (kn(-5,-7)) shows with 100% certainty which knot-numbers are not primes (erased bars from the kn pattern) and “where” (bars) one have to search for a prime number. However, if you want the final pattern (bars that are 100% primes), you will have to consider the contributions of the kc(11), kc(13), kc(17) and kc(19). (Figure 2) There is no need to considerer multiples of 23, 25, 29, 31, etc. because their first contributions are already too big for the desired interval (5 to 469) as follows: 232 = 529, 252 = 625 and, so on. Also, lets point to the fact that only non composite knot-numbers (simple knot-numbers or prime numbers) will contribute with new multiples to be discarded. As long as you move forward to bigger numbers it’ll increase the number of kc numbers and decrease the numbers of ks numbers (primes) around the knots. This means that the pattern tends to a nearly steady final (but never completely steady, because primes are infinite) pattern with a huge period Tn.

4 FIGURE 2 –Bar-code pattern for kn & ks (100% PRIMES) numbers

ks(prime)

kn(-5,-7)

kn

5

25

49

259

469

Figure 2. From the bottom to the top: (a) kn numbers (two for every knot); (b) kn(-5,-7) numbers after subtracting the kc(5) and kc(7) contributions and (c) ks numbers (primes) after subtracting the kc(11), kc(13), kc(17) and kc(19) contributions. Here one has 100% primes.

FIGURE 3 – Number of PRIMES for every thousand

Number of primes 200

y = -8.1863Ln(x) + 182.66 160

120

80

40

0 0E+00

1E+05

2E+05

3E+05

4E+05

5E+05

6E+05

Natural numbers

Figure 3. From zero to 50,000 the number of primes quickly decreases from 333 to 100 for every thousand. Approximately 70% of the kn vanishes, remaining only 30% to be tested. The period of ks pattern is already huge at this point.

To work with big numbers doesn’t imply in consider “all” the kc contributions . The smaller kn are the ones that contributes the most to clean up the final pattern because their periods are also small. For every 30 natural numbers there are 2 contributions due to the 5 (sweeping up 2 kc); while, for example, the number 19 help us only at every 114 natural numbers, an so on for bigger numbers. The prime numbers pattern itself (ks final pattern) is a combination of patterns and this is why it “looks” randomly. It’s similar to a combination of several waves with the same amplitude but different wavelengths: the final result will be a package with a huge wavelength. Also, concerning the prime numbers pattern, lets remark that its period is always changing and growing, it’s always in movement. At the number 25 the kn(-5) pattern start to run from 25 to the infinity, however it can be easily

5 identified only from 25 to 49. From 49 to ahead the contributions of the number seven starts to run also and the pattern changes, again. From 25 to 49 one can see a little peace of the kn(-5) pattern (as the small peak of an iceberg above water). But from 49 ahead a new pattern starts to run, it’s the kn(-5,-7) pattern! From 49 to 121 one will be able to identify the kn(-5,-7) pattern solely, but at 121 starts the kn(-5,-7,-11) pattern with its new period and so on, to the infinity. For unaware eyes it may seem that there is no pattern in there. “They appear among the integers seemingly at random, and yet not quite: There seems to be some order or pattern, just a little below the surface, just a little out of reach." From: Underwood Dudley, Elementary Number Theory (Freeman, 1978). But the pattern is there. Not a steady pattern (not a picture) but a moving pattern (as in a movie). To map all the prime numbers among the natural numbers is not a hard task from the theoretical point of view, but it’s something that will take some time, due to the computational work when dealing with huge numbers. A good deal of the small primes are already mapped, but till now, no one was able to use the knowledge about the distribution of the smaller primes to predict the biggest ones, only because, no one was able to “project” these results to the infinity! And, to do so, all one needs is to be able to find the patterns and to predict its periods. To generate a huge pattern (as clean as possible) and to project it to the infinity is a task that will take some effort and time (from the computational point of view) but it’s something that one will has to do only once, because the pattern will work forever! Remember, all patterns you construct for small knot-numbers, will work for the big ones two, giving you a 100% certainty about the blanks in the plot (the kn that are discarded, or else, the kc contributions) and a 50%, 70%, ...100% (depending on how many kc you considered) for the bar-points, ks (primes). Now, lets stop and pay some attention to the following: every time you subtract the contribution of the multiples of a knot-number you are cleaning up the kn pattern from 5 to the infinite! For every thousand natural numbers there are approximately 333 kn numbers, but the number of primes (ks) quickly decreases while the magnitude of the natural number increases. At 50,000 approximately 70% of the knot-numbers already vanished and the pattern will seem less crowded with only 30% of the initial lines. However, the period of such a pattern will be huge!

FIGURE 4 –Fabric pattern for KN & KC numbers

Figure 4. From the bottom to the top: (a) kn numbers; (b) kc(5) contributions ; (c) kn(-5) numbers after subtracting the kc(5) contributions; (d) kc(7) contributions; (e) kn(-5,-7) numbers after subtracting the kc(5) and kc(7) contributions.

Let’s remark, also, that there is no need of previous information about the primes before the interval you are interested, you need only the information about the knot-numbers and they can be easily found. Also, you can focus on any interval you want: from 5 to 469 or, for example, from ( 6 × 2123456789 ± 1 ) to ( 6 × 2 987654321 ± 1 ) ! It doesn’t matter, because, you only need to discard the kc numbers at this interval. But the amazing thing is that to save time and effort, all you have to do is to project the pattern (Figure 1) through the interval you are interested. Lets say, based on Figure 1, one can predict, with 100% accuracy, without no calculations, that the number

6 91 + 42 × 2123456789 is located around a knot and is not a prime! The same thing, for example, for 91 + 210 × 2123456789 . But, not of course, for 91 + 30 × 2123456789 , because the number 91 is a multiple of 7 (not 5). Finely, let’s play a little with the patterns we have shown previously (Figure 1), in order to make them more fun. To do so, let’s change the markers to big pink exes and see how it looks. Figure 4 shows the same data as Figure 1, but now the plots look more like a fabric (instead of a bar code) with a characteristic pattern for the threads. As long as you successively pull off the kc(5) and kc(7) threads, new patterns show up (patterns with threads more loosely woven). A new pattern appears in front of our eyes every time we pull of the contributions of the next set of kc numbers! There is beauty in theses patterns! And, as long as prime numbers are natural numbers, it’s logic to expect finding these patterns in nature. Everything that grows sequentially (1+1, 2+1, etc.) should be able, at some level, to express these beautiful patterns shown in Figure 4. The patterns are, indeed, so artistic that they can be used as a t-shirt print! An artistic proof of the fact that: prime numbers do have patterns!!! Finely let us finish with two golden quotes from The music of primes (Du Sautoy, M., 2003): (1) "The primes have tantalized mathematicians since the Greeks, because they appear to be somewhat randomly distributed but not completely so." (T. Gowers, Mathematics: A Very Short Introduction, Oxford Univ. Press, 2002, p.118). Exactly! It seems randomly distributed but it isn’t. (2) "There are two facts about the distribution of prime numbers which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that despite their simple definition and role as the building blocks of the natural numbers, the prime numbers... 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." (Don Zagier, Bonn University inaugural lecture). I could not express myself better!

CONCLUSION

Note that we have started the knot numbers sequence at the number 5, but the real first knot is at the number zero (6-6=0), what makes the number 1 the first knot number (a knot number on the right). At the opposite side we have the reflection in the mirror: the negative knot-numbers. Note, again, that a prime number is always a knot number for all number from 5 to infinity, there is no exception! This means that the number 2 and the 3 are from different categories and cannot be called as prime numbers. Indeed they can be divided only by one and their selves, but only because they are two small to have multiples. They are construct numbers as the number one, but of second and third class. Finely, prime numbers do obey laws (very simple, indeed), do follow patterns (very predictable) and can be completely explained, understood and predicted. Prime numbers do look random at first sight, but they are not random at all. They are puzzling, stunning, perplexing, magic, etc. but, cannot be described as a mystery.

REFERENCES Duha, J. Prime numbers without mystery, Lecture at the Department of Mathematics of the Universidade Federal do Parana - UFPR, Curitiba, Brasil, 2006. Duha, J. Prime numbers without mystery, The Mathematics Preprint Server, www.mathpreprints.com, 2004. Duha, J. Prime numbers as potential pseudo-random code for GPS signals, Boletim de Ciencias Geodesicas, v.10, p.215-224, 2004b. Du Sautoy, M. The music of the primes, 2003.