Prime Numbers Without Mystery - Final Solution

1 PRIME NUMBERS WITHOUT MYSTERY FINAL SOLUTION Jânia Duha ABSTRACT To try understand the rule of prime numbers on the scenario of natural numbers is one of the most interesting tasks in mathematics today. In this work we present a model that shows how prime numbers appear in the natural sequence 1, 2, 3, ..., where they are hidden, and how to find then with 100% certainty. We also show that prime numbers should not be known as "random numbers", because they are not random at all. In fact, they follow a set of periodic patterns that can be easily found and understood.

INTRODUCTION

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. The first number is one, the second is 1+1=2, the third is 2+1=3, and so on: 1, 2, 3, 4, 5, 6, 7,8, 9, 10,11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,...

The number one is a construct number. By adding one successively you construct all the sequence. But the second and the third number can provide a good part of the sequence too. With the two we have (in bolt) 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 ... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

With the three we have: 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 ... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 3 6 9 12 15 18 21 24 27 30

But, note that some numbers are provided by the two and the three simultaneously. We will call these repeated numbers (6, 12, 18, 24, 30…) as the "knots" of the sequence (in shadow): 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 ... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 3 6 9 12 15 18 21 24 27 30

Note also, that these knots have always a number on the left and a number on the right side that cannot be provided by the two or the three. Let us call this special numbers as “prime numbers”. If you look more carefully you will note that some of these numbers are pure (can be divided only by 1 and by their selves) and some are composite, or else, they can be divided also by other numbers (but, only prime numbers as defined in this work: numbers on the left and right hand sides of the knots). So, the numbers around the knots can be “pure prime numbers” (pp) or “composite prime numbers” (cp).

2 The cp numbers are composed by the multiplication of prime 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 prime numbers. Table 1 – Composite prime numbers. 5

7

11

25

5

13

17

55

95 133

77 121

119 187

221 289

145

30 10-20

42 28-14

203 319

66 22-44

275 325

78

299 391 323 361

19

Periodicity

125 175

209 247

17

29

161 253

143 169

13

25

115

65 91

11

23

85

35 49

7

19

52-26

377 493

102 34-68

425 475

114

437

76-38

551

These composite primes are very easy to calculate and it is also very easy to predict their behavior through the infinity because they present a clear periodicity as shown in the Table 1 The sequence of prime numbers and theirs knots are even more easy to calculate 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. You can be 100% sure that on the left and on the right side of a knot you will find a prime number. But if you want a “pure prime” number so, what you need to do it’s only identify the “composite primes” and eliminate them. All that will remain is the pure primes that you are seeking “prime numbers” = “pure primes” + “composite primes” so “pure primes” = “prime numbers” - “composite primes” So, now we point to the fact that if the primes and the composite primes have a predictable behavior (with clear periodicities), the "pure primes" should be predictable, too. Next, we will show the knots (shadow), the "pc" numbers (bolt) and the "pp" numbers (bolt underlined) form 5 to 100: 5 29 53 77

6 30 54 78

7 31 55 79

8 32 56 80

9 33 57 81

10 34 58 82

11 35 59 83

12 36 60 84

13 37 61 85

14 38 62 86

15 39 63 87

16 40 64 88

17 41 65 89

18 42 66 90

19 43 67 91

20 44 68 92

21 45 69 93

22 46 70 94

23 47 71 95

24 48 72 96

25 49 73 97

26 50 74 98

27 51 75 99

28 52 76 100

To obtain all possible pure primes from one to a number as big as one can possible image, you have only to follow the simple rules described above. One by one, the pure primes will appear with 100% of accuracy without the need of any proof of primarility. THE EQUATIONS 1. Prime numbers - "p"

3

Prime numbers, as defined above in this work p1 = 5 , p2 = 7 , p3 = 11 , p4 = 13 , p5 = 17 , p6 = 19, p7 = 23, p8 = 25, ∞,

can be obtained through the following simple relation: p = 6k ±1 , with k = 1, 2, 3, 4.... , or else p 2 k = 6k + 1 ,

for the primes on the right side, and p 2 k −1 = 6k − 1

for the primes on the left side.

2. Composite prime numbers – "cp" Multiplying all possible primes we obtain: cp 1n = p 1 . p n

with with with

cp 2 n = p 2 . p n cp 3n = p 3 . p n

M

n = 1, 2, 3, 4, 5, 6, 7, ..... n = 2, 3, 4, 5, 6, 7, ..... n = 3, 4, 5, 6, 7, .....

M

Examples:

cp 11 = cp 12 = cp 13 =

p1 . p1 p1 . p2 p1 . p3

= = =

5 . 5 = 25 5 . 7 = 35 5 . 11 = 55

cp 22 = cp 23 = cp 24 =

p2 . p2 p2 . p3 p2 . p4

= = =

7 . 7 = 49 7 . 11 = 77 7 . 13 = 91

cp 33 = cp 34 = cp 35 =

p3 . p3 p3 . p4 p3 . p5

= = =

11 . 11 11 . 13 11 . 17

or

or = 121 = 143 = 187

Generalizing this relations 2.1 Left-Prime & Left-Prime cp m , n = (6k − 1) * (6l − 1) ,

k = 1, 2, 3, 4, ... ,

l = k, k+1, k+2 ... , m = 2k-1, n = 2l-1

2.2 Left Prime & Right-Prime cp m , n = (6k − 1) * (6l + 1) ,

k = 1, 2, 3, 4, ... ,

l = k, k+1, k+2, …, m = 2k-1, n = 2l

2.3 Right-Prime & Left-Prime cp m,n = (6k + 1)* (6l − 1)

,

k = 1, 2, 3, 4, ... , l = k+1, k+2, k+3, ..., m = 2k, n = 2l-1

2.4 Right-Prime & Right-Prime cp m,n = (3k + 7 )* (3l + 7 ) ,

k = 0, 2, 4, 6, ... (m = k + 2), l = k, k+2, k+4, …, m = 2k, n = 2l

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2. Pure prime numbers - pp Pure prime numbers are all the primes as defined above except the composite primes. In order to calculate all possible pp numbers from 5 to 3,789,547, or any number as big as one can imagine, one by one, with 100% of accuracy and no need of any proof of primarility, first, we compute the p numbers, second, we compute de cp numbers and subtract them from the set of original p numbers. Some mathematical concepts are so simple, that, for example, if someone postulates that: “If you add 1 to an even number the result will be an odd number, no matter how big is the number”. The theory above is like that, so simple, that we can be completely sure that it will work well for big primes as well as for the smaller ones. DISCUSSION The pattern of the prime numbers is very simple and presents one easy-to-see pattern as shown in the Figure 1, where the type (1) (first set from the bottom) is the prime numbers with the knots between the left and right primes. The type (2) is the composite primes that must be discarded, and the type (3) is the pure primes (on the top). The period tic behavior of the pure primes is not so easy to see, but it’s the result of two other periodic patterns: the p and cp. While the period of p numbers is 6 for all intervals you consider, the period of the cp numbers depends on the maximum value you are considering (is this case: 300). Figure 1 Prime numbers distribution from 5 to 300.

6

(1) p, (2) cp, (3) pp 5

Type (p, cp, pp)

4

3

2

1

0 0

50

100

150

200

250

300

Prime Numbers

To understand the periodicity of cp numbers is important because if you have this information you can understand and predict the periodicity of the pure prime numbers. Let us begin with small numbers and analyze a distribution from 5 to 48, for example. This distribution has only one number contributing to the set of cp numbers that must be discarded: the 5, because the first contribution of the 7 is 7x7=49, the 11 is 11x11=121, and so on. So we have the following numbers that are cp primes: 5x5=25 and 5x7=35 as shown in the Figure 2. Figure 2 Prime numbers distribution from 5 to 48.

5

6

(1) p, (2) cp, (3) pp

5

Type (p, cp, pp)

4

3

2

1

0 0

10

20

30

40

Prime Numbers

In the Figure 1 and 2 the type (3) are pure primes with 100% accuracy. And, if you take into account all the numbers that generates the set of cp primes that appears up to the number you want achieve, e.g., 300 in Figure 1 and 48 in Figure 2, you will always have 100% of accuracy. However, when talking about large numbers it may be reasonable to take into account only the first cp primes in order to maintain the periodicity easy to control. Because, the more numbers you take into account to construct the cp primes, bigger will be the period that will rule the pp behavior. Lets say that you don't want much work so, to predict the position of pp primes from 5 to 100 you, will take into account only the five: 5x5=25, 5x7=35, 5x11=55, 5x13=65, 5x17=85 and 5x19=95. While you should take into account also the contribution of the seven: 7x7=49, 7x11=77, 7x13=91. Figure 3 Prime numbers distribution from 5 to 100.

6

(1) p, (2) cp, (3) pp 5

Type (p, cp, pp)

4

3

2

1

0 0

20

40

60

80

100

Prime Numbers

In this case, we don't have 100% of accuracy, but we can find a pattern for the pure primes with period Tpp = 30 (Tp = 6 , Tcp = 30). This approximate approach will work even better for big numbers, because the great majority of composite primes yields from the smallest primes of the sequence.

6

There are approximately 333 primes for every thousand, but the number of pure primes decreases exponentially. The first thousand (5 to 1005) has 166 pure primes; the second thousand (1005 to 2005) has 136 pure primes, and so on. The Figure 4 shows how the number of pure primes decreases for every thousand you consider. Figure 4 Number of Pure Primes - PP

180

Number of Pure Primes

160

140

120

100

80

60 0

20000

40000

60000

80000

100000

120000

Natural Numbers

All prime numbers around the knots are possible pure primes. This means that we have 333 possible pure primes for every thousand. But when you start to cut of the composite primes this number decreases quickly. After 120 thousands (120000) there are, only, 70 possible primes to be considered (20% of the initial number). And if you let your computer work a little more, this number will decrease even more. It's important to remark that, if you want to work, e.g., with numbers between 2 x 101234567 and 2 x 102234567 you do not need to calculate all composite primes before that. You will achieve very accurate results if you consider only the contribution of a set of composite primes big enough to minimize the number of "possible" pure primes. If you stop to consider the composite primes at some point, this will provide you a pattern for the pure primes, that will repeat it self through the infinity with periodicity T , and that will point to the few numbers that are able to be pure primes. Of course that, if you do have time and a good computer you can compute the entire composite primes that have to be discarded and so, the numbers that will remain will be pure primes with 100% certainty. The program to calculate pure primes is very simple and short (one page) because the equations are simple. There is no need of dealing with complex numbers or non-trivial mathematical hypothesis. It's really hard to understand why pure primes are known as "random numbers", because they are not random at all. In fact, they follow a set of periodic patterns that can be easily found and understood. It seems that the problem around pure primes is not a mathematical problem (not with the model we are proposing here) but a computational one where, what we have to find is "how make computers work easily and faster when concerning big numbers".