Download Periodical Appearance of Prime Numbers

September 6, 2016
Periodical Appearance of Prime Numbers
Yoshiki Kaneoke
Department of System Neurophysiology, Graduate School, Wakayama Medical University
Highlights
1. When prime number (Pi, i=1,2,3,,,) is treated as radian, plot of (i, sin(Pi)) seems to make 20
sigmoidal lines. Exception is the 5th prime number 11, which is outside of these lines.
2. These lines seem to continue up to 99,999,999,977 investigated, and the periodicity increases as
the prime number increase.
3. Three dimensional plot of (sin(Pi), cos(Pi), i) and two dimensional plot of (i, Di) (Di is the
remainder after division of 180*Pi/πby 360) shows that the 20 lines are divided into two groups
with a gap. Each group has 10 lines.
4. The value of angle (that is Pi radian or Di degrees) tends to cluster to make a dotted line with a
specific pattern.
5. It is hypothesized that all but one (11) prime numbers are on the 20 lines and could have two
different properties. It is possible, however, that several prime numbers other than 11 might be
found within a gap between the two groups’ lines.
Acknowledgments
1. Prime numbers used were downloaded from the Websites:
http://www.hyogo-c.ed.jp/~meihoku-hs/club/astronomy-p.html
http://compoasso.free.fr/primelistweb/page/prime/liste_online_en.php
2. Visualization was done with MATLAB 2015a.
First, the results for the first 2,000 (2 to 17389) prime numbers are shown in Figure 1 - 3. Second, the
results are shown for the 2,000 consecutive prime numbers starting at various prime numbers. The
sigmoidal lines on the plot of (i, sin(Pi)) are getting difficult to see for the larger prime numbers plots.
But the plot of (i, Di) shows clear lines and gaps. The easiest way to check the fact that the dots are
on the 20 lines is as follows: Print the right side of the Figure 2. Cut out the figure and make a cylinder
by putting the bottom (x-axis) and the top of the figure together perfectly. The printed surface is
outside, of course. It is unknown what function can draw these lines. The findings seem to be true for
the several sets of the 2,000 prime numbers investigated, but not every prime numbers below
99,999,999,977 could be tested. The prime numbers used were not validated by the authorized
programs.
September 6, 2016
Figure 1. Plot of (i, sin(Pi)), i=1,2,3,, , 2000.
All dots but the dot for 11 (the plot is shown in the red circle) are on the sigmoidal lines.
2 - 17 389
1
0 .8
0 .6
0 .4
0 .2
0
- 0 .2
- 0 .4
- 0 .6
- 0 .8
-1
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Figure 2. Three dimensional plot of (sin(Pi), cos(Pi), i) (left) and two dimensional plot of
(i, Di) (right)
for the first 2,000 prime numbers (2 – 17389).
The dots make obvious 20 lines except the dot for 11 shown in the red circle. Note the gap making two
groups of lines, each of which consists of 10 lines.
One group of 10 lines
Other group of 10 lines
2 - 1 7389
2 - 173 89
350
300
2000
1800
250
1600
1400
1200
200
1000
800
150
600
400
200
100
0
1
0. 5
1
50
0 .5
0
0
-0 . 5
- 0 .5
-1
-1
0
0
200
400
600
800
1000
1200
1400
1600
1800
2000
September 6, 2016
Figure 3.
Three dimensional plot (sin(Pi), cos(Pi), i) shown at different angles as Fig. 2.
The angle of Pi (that is, Pi radian or Di degrees) tends to be similar as shown by the gaps (some are
indicated by the red arrows). Generally, the dot line shown at the right has a specific pattern: 4
consecutive dots with a gap.
2 - 173 89
2 - 173 89
1
0. 8
0. 6
2000
1000
0
0. 4
0. 2
1
0. 8
0
0. 6
0. 4
-0 . 2
1
0. 2
0 .8
0
0 .4
-0 . 2
-0 . 6
0 .2
-0 . 4
0
- 0 .2
-0 . 6
-0 . 8
- 0 .4
- 0 .6
- 0 .8
-1
- 0 .8
-1
Figure 4.
-0 . 4
0 .6
-1
-1
-0 . 8
- 0 .6
- 0 .4
- 0 .2
0
0 .2
0 .4
The same pattern is seen for the 2,000 prime numbers from 81817 to 104729.
The plots are the same as Fig. 1-3.
818 17 - 1 0472 9
81 817 - 104 729
350
1
0 .8
300
0 .6
250
0 .4
0 .2
200
0
150
- 0 .2
- 0 .4
100
- 0 .6
50
- 0 .8
-1
0
200
0 .6
400
600
800
1000
1200
1400
1600
1800
2000
0
0
200
400
600
800
8 181 7 - 10 472 9
1000
1200
1400
1600
1800
8 1817 - 10 4729
1
0. 8
2000
0. 6
1800
1600
0. 4
1400
1200
0. 2
1000
0
800
600
-0 . 2
400
-0 . 4
200
0
1
-0 . 6
1
0. 5
0 .5
0
0
-0 . 5
- 0 .5
-1
-1
-0 . 8
-1
-1
- 0 .8
- 0 .6
- 0 .4
- 0 .2
0
0 .2
0 .4
0 .6
0 .8
1
2000
0 .8
1
September 6, 2016
Figure 5.
The 20 lines seem to continue up to 99,999,999,977.
The frequency of the sigmoidal line increases as the order increase as shown by the angle of the line
(indicated by the red lines on the gaps, just to assist checking the line angle. Note that the gaps and
the lines made by the dots are not straight line.). From the top, plots (i, Di) for the 2,000 prime
numbers from 2 to 17,389, from 81,817 to 104,729, from 7,471,771 to 7,503,577, from 999,958,577 to
999,999,937, from 99,999,950,447 to 99,999,999,977 are shown.
2 - 1 7 3 8 9
3 5 0
3 0 0
2 5 0
2 0 0
1 5 0
1 0 0
5 0
0
0
2 00
4 0 0
60 0
8 0 0
1 00 0
1 2 0 0
1 40 0
1 6 00
18 0 0
2 00 0
1 2 0 0
1 40 0
1 6 00
18 0 0
2 00 0
1 40 0
1 6 00
18 0 0
2 00 0
1 40 0
1 6 00
18 0 0
2 0 00
1 40 0
1 6 00
18 0 0
2 00 0
8 1 8 1 7 - 1 0 4 7 2 9
3 5 0
3 0 0
2 5 0
2 0 0
1 5 0
1 0 0
5 0
0
0
2 00
4 0 0
60 0
8 0 0
1 00 0
7 4 7 1 7 7 1 - 7 5 0 3 5 7 7
3 5 0
3 0 0
2 5 0
2 0 0
1 5 0
1 0 0
5 0
0
0
2 00
4 0 0
60 0
8 0 0
1 00 0
1 2 0 0
9 9 9 9 5 8 5 7 7 - 9 9 9 9 9 9 9 3 7
3 5 0
3 0 0
2 5 0
2 0 0
1 5 0
1 0 0
5 0
0
0
2 00
4 0 0
60 0
8 0 0
1 00 0
1 2 0 0
9 9 9 9 9 9 5 0 4 4 7 - 9 9 9 9 9 9 9 9 9 7 7
3 5 0
3 0 0
2 5 0
2 0 0
1 5 0
1 0 0
5 0
0
0
2 00
4 0 0
60 0
8 0 0
1 00 0
1 2 0 0
September 6, 2016
Figure 6.
The specific dots pattern is also seen up to 99,999,999,977 investigated.
The pattern means that the prime number angle (Di) tends to gather within restricted small ranges
when checked with a small consecutive number of prime numbers like 2,000.
74 717 71 - 7 5035 77
9999 585 77 - 9 9999 993 7
1
1
0 .8
0 .8
0 .6
0 .6
0 .4
0 .4
0 .2
0 .2
0
0
- 0 .2
- 0. 2
- 0 .4
- 0. 4
- 0 .6
- 0. 6
- 0 .8
- 0. 8
-1
-1
-1
- 0 .8
- 0 .6
- 0 .4
- 0 .2
0
0 .2
0 .4
0 .6
0 .8
1
-1
-0 . 8
-0 . 6
-0 . 4
-0 . 2
0
0 .2
0 .4
0 .6
0 .8
1
0. 6
0. 8
1
99 9999 504 47 - 9 999 9999 977
1
999 99950 447 - 9 99999 99977
0 .8
0 .6
0 .4
0 .2
2 00 0
1 00 0
0
0
1
0.8
1
0.6
- 0 .2
0.8
0 .4
0 .6
0 .4
0 .2
- 0 .4
0.2
0
0
-0.2
- 0 .6
- 0. 2
- 0 .4
-0 . 4
- 0. 6
- 0 .8
-0.6
- 0. 8
- 0 .8
-1
-1
-1
-1
- 0 .8
- 0 .6
- 0 .4
- 0. 2
0
0 .2
0 .4
September 6, 2016
Figure 7.
Model of the prime numbers periodicity.
Radian angle of the prime number seems to be on the 20 lines, which could be divided into two groups
(illustrated below by the cyan and the magenta lines). Each group has 10 lines and is separated by
the gap of one line. One prime number 11 is found in the gap. These lines continue increasing the
periodicity. As shown below, the space between the lines are getting smaller as the prime number
increases.
18
16
14
12
10
8
6
4
2
1
05
.
0
1
0
08
.
06
.
04
.
02
.
0
-02
. -04
.
-05
.
-06
. -08
.
-1
-1
September 6, 2016
Appendix
Matlab code for the plots.
%%
P=primes(17390); %first 2000 prime numbers
I=1:2000;
D=2000;
figure
plot3(sin(P(D-1999:D)),cos(P(D-1999:D)),I,'.')
title([num2str(P(D-1999)),'-',num2str(P(D))])
figure
plot(sin(P(D-1999:D)),cos(P(D-1999:D)),'.')
title([num2str(P(D-1999)),'-',num2str(P(D))])
axis([-1.1 1.1 -1.1 1.1])
axis square
figure
plot(sin(P(D-1999:D)),'.')
title([num2str(P(D-1999)),'-',num2str(P(D))])
axis([0 2001 -1.1 1.1])
figure
plot(mod(360*P(D-1999:D)/(2*pi),360),'.')
title([num2str(P(D-1999)),'-',num2str(P(D))])
axis([0 2001 0 360])
%%