art theory 101~ Butterfly Primes

Butterfly Primes
~let the beauty seep in~

Copyright 2005, Reginald Brooks. All rights reserved.

Date completed: 12-07-05

 

Introduction

The lure and elusiveness of prime numbers, those very same numbers which all the other natural whole numbers can be reduced to, is thousands of years in the making, captivating the minds of anyone...perhaps everyone...who has ever contemplated their beauty and mystery. Can this beauty reveal a visual and mathematical pattern?

Given Prime Definition

A prime number ( P ) is any natural whole number ( N ), greater than one, which is reducible only by itself and one.

Postulate 1: "Axis vs. Matrix, and Unique Squares"

On a Multiplication Matrix (Table) where all natural whole numbers 1,2,3,...are placed on the x- and y-axis with 1 at the origin, it follows that (Fig 1, Table I):

a. All primes, and only primes, remain when all the matrix numbers ( NM ) are eliminated from all the axial (x- or y-axis) numbers, as:

P = Nx-axis - NM                   (1)

b. Similarly, only primes remain when the product of all x-axis numbers times the y-axis numbers are eliminated from all numbers, as:

P = N-(Nx-axis x N y-axis)         (2)

c. And again, as all primes are odd (except P=2=P2 ), only primes remain when all odd matrix numbers are eliminated from all the odd numbers (plus N2=2 ), as:

P = (Nodd-x-axis + N2)-NM-odd       (3)

d. Not only are the squares of all primes (except P2 ) odd, and (except P5 ) end in either 1 or 9, the squares of the squares all end in 1, as:

P2except P2 = Nodd-ends in 1 or 9    (4)

(P2)2 = Nodd-ends in 1               (5)

e. The squares of all primes are the only unique numbers on the matrix, as:

P2 = NM-unique .                 (6)

 

 

Fig 1:Postulate 1: 'Axis vs. Matrix, and Unique Squares

Fig. 1
(click to enlarge image)
Best viewed 800x600, F-11 key

Table I

   Table I. Postulate 1

     All numbers,      All Matrix numbers,   All Prime numbers,
     N x-axis       -    NM                  = P
     2-50...           4-50...               from 1-50...
     as a group        as a group            as a group
    ____________       ___________________   __________________
      -      		4			
      2        		6			2
      3           	8			3
      4     		9			
      5      		10			5
      6    		12			
      7       		14			7			
      8    		15
      9    		16
     10    		18
     11        		20			11
     12    		21
     13      		22			13
     14    		24
     15    		25			
     16    		26
     17        		27			17
     18    		28
     19        		30			19
     20    		32
     21    		33
     22    		34
     23        		35			23
     24    		36
     25    		38
     26    		39
     27    		40
     28    		42
     29    		44			29
     30    		45
     31    		46			31
     32    		48
     33    		49
     34    		50
     35    
     36    
     37    					37   
     38    
     39    
     40    
     41    					41     
     42    
     43       
     44    
     45    
     46    
     47    					47  
     48    
     49    
     50      

Commentary

All natural whole numbers, N = 1, 2, 3,... , placed on the x- and y-axis, with the unit number 1 at the origin to form the Multiplication Matrix (Table). The Multiplication Matrix Numbers, NM , equals the grid of products formed from multiplying the Nx-axis and Ny-axis (excluding N=1 ) numbers. All prime numbers, P , are only found, by definition, on the x- or y-axis and are not part of the matrix numbers, NM . Removing (not actually subtracting) all the numbers from this axis which also appear in the matrix table will leave only the prime numbers. The corollary to this is that the squares of the primes are the only unique numbers within the matrix table. Fig.1.

Postulate 2: "Divisible Differences Predict Primes"

a. Except for P2,3,5 the difference in the squares of any prime is divisible by 24, as (Fig.2, Table II):

(P>)2 - (P<)2 = n24                (7)

[(P>)2 - (P<)2] / 24 = n           (8)

[(P>)2 - (P<)2] / n = 24     v      (9)

where P> = greater, P7 or higher, and, P< = lesser

b. The square root of any squared prime plus a multiple of 24 (n24) equals the next prime, as:

Pnext = [(P<)2 + n24]½             (10)

where P< = P5 or higher.

Fig 2:Postulate 2: 'Divisible Differences Predict Next Primes'

Fig. 2
(click to enlarge image)
Best viewed 800x600, F-11 key

Table II

     Table II Postulate 2 

    All                   (a.)                                 (b.)
  Primes,                                    |
    P     squared  (P>)2 - (P<)2   =    n24   |   n24+(P<)2    = P2next    (P2next)1/2   =  P
   ____            ______________    _____   |  __________    _______    __________  _____
     2                                       |          
     3                                       |
     5    ^2 =    25	25 -9 =		     |	       
     7    ^2 =    49	49 -25 =      (1)24  |  (1)24 +  25   =	   49      49	         7
    11    ^2 =   121	121 -49 =     (3)24  |  (3)24 +  49   =	  121	  121	        11
    13    ^2 =   169	169 -121 =    (2)24  |  (2)24 +  121  =   169     169 	        13
    17    ^2 =   289	289 -169 =    (5)24  |  (5)24 +  169  =   289     289           17
    19    ^2 =   361	361 -289 =    (3)24  |  (3)24 +  289  =   361     361           19
    23    ^2 =   529	529 -361 =    (7)24  |  (7)24 +  361  =   529     529           23
    29    ^2 =   841	841 -529 =   (13)24  | (13)24 +  529  =   841     841           29
    31    ^2 =   961	961 -841 =    (5)24  |  (5)24 +  841  =   961     961           31
    37    ^2 =  1369	1369-961 =   (17)24  | (17)24 +  961  =  1369    1369           37
    41    ^2 =  1681	1681-1369=   (13)24  | (13)24 + 1369  =  1681    1681           41
    43    ^2 =  1849	1849-1681=    (7)24  |  (7)24 + 1681  =  1849    1849           43
    47    ^2 =  2209	2209-1849=   (15)24  | (15)24 + 1849  =  2209    2209           47
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    53    ^2 =  2809	2809-2209=   (25)24  | (25)24 + 2209  =  2809    2809           53
    59    ^2 =  3481	3481-2809=   (28)24  | (28)24 + 2809  =  3481    3481           59
    61    ^2 =  3721	3721-3481=   (10)24  | (10)24 + 3481  =  3721    3721           61
    67    ^2 =  4489	4489-3721=   (32)24  | (32)24 + 3721  =  4489    4489           67
    71    ^2 =  5041	5041-4489=   (23)24  | (23)24 + 4489  =  5041    5041           71
    73    ^2 =  5329	5329-5041=   (12)24  | (12)24 + 5041  =  5329    5329           73
    79    ^2 =  6241	6241-5329=   (38)24  | (38)24 + 5329  =  6241    6241           79
    83    ^2 =  6889	6889-6241=   (27)24  | (27)24 + 6241  =  6889    6889           83
    89    ^2 =  7921	7921-6889=   (43)24  | (43)24 + 6889  =  7921    7921           89
    97    ^2 =  9409	9409-7921=   (62)24  | (62)24 + 7921  =  9409    9409           97
   ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   101    ^2 = 10201  10201-9409=    (33)24  | (33)24 + 9409  = 10201   10201          101
   103    ^2 = 10609  10609-10201=   (17)24  | (17)24 +10201  = 10609   10609          103
   107    ^2 = 11449  11449-10609=   (35)24  | (35)24 +10609  = 11449   11449          107
   109    ^2 = 11881  11881-11449=   (18)24  | (18)24 +11449  = 11881   11881          109
   113    ^2 = 12769  12769-11881=   (37)24  | (37)24 +11881  = 12769   12769          113
   127    ^2 = 16129  16129-12769=  (140)24  |(140)24 +12769  = 16129   16129          127
   131    ^2 = 17161  17161-16129=   (43)24  | (43)24 +16129  = 17161   17161          131
   137    ^2 = 18769  18769-17161=   (67)24  | (67)24 +17161  = 18769   18769          137
   139    ^2 = 19321  19321-18769=   (23)24  | (23)24 +18769  = 19321   19321          139
   149    ^2 = 22201  22201-19321=  (120)24  |(120)24 +19321  = 22201   22201          149
   ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   151    ^2 = 22801  22801-22201=   (25)24  | (25)24 +22201  = 22801   22801          151
   157    ^2 = 24649  24649-22801=   (77)24  | (77)24 +22801  = 24649   24649          157
   163    ^2 = 26569  26569-24649=   (80)24  | (80)24 +24649  = 26569   26569          163
   167    ^2 = 27889  27889-26569=   (55)24  | (55)24 +26569  = 27889   27889          167
   173    ^2 = 29929  29929-27889=   (85)24  | (85)24 +27889  = 29929   29929          173
   179    ^2 = 32041  32041-29929=   (88)24  | (88)24 +29929  = 32041   32041          179
   181    ^2 = 32761  32761-32041=   (30)24  | (30)24 +32041  = 32761   32761          181
   191    ^2 = 36481  36481-32761=  (155)24  |(155)24 +32761  = 36481   36481          191
   193    ^2 = 37249  37249-36481=   (32)24  | (32)24 +36481  = 37249   37249          193
   197    ^2 = 38809  38809-37249=   (65)24  | (65)24 +37249  = 38809   38809          197
   199    ^2 = 39601  39601-38809=   (33)24  | (33)24 +38809  = 39601   39601          199
   ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   211    ^2 = 44521  44521-39601=  (205)24  |(205)24 +39601  = 44521   44521          211
   223    ^2 = 49729  49729-44521=  (217)24  |(217)24 +44521  = 49729   49729          223
   227    ^2 = 51529  51529-49729=   (75)24  | (75)24 +49729  = 51529   51529          227
     
    ~~~~~~~~~random clusters hereafter  through the first 10K primes~~~~~~~~~~~           
   ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   283    ^2 =  80089                        |
   293    ^2 =  85849  85849-80089= (240)24  |(240)24 + 80089  =  85849    85849       293
   307    ^2 =  94249  94249-85849= (350)24  |(350)24 + 85849  =  94249    94249       307
                                             |
   449    ^2 = 201601                        |
   457    ^2 = 208849 208849-201601=(302)24  |(302)24 +201601  = 208849   208849       457
   461    ^2 = 212521 212521-208849=(153)24  |(153)24 +208849  = 212521   212521       461
                       			     |
   617    ^2 = 380689    		     |
   619    ^2 = 383161 383161-380689=(103)24  |(103)24 +380689  = 383161   383161       619
   631    ^2 = 398161 398161-383161=(625)24  |(625)24 +383161  = 398161   398161       631
    					     |
   787    ^2 = 619369  		             |   
   797    ^2 = 635209 635209-619369=(660)24  |(660)24 +619369  = 635209   635209       797
   809    ^2 = 654481 654481-635209=(803)24  |(803)24 +635209  = 654481   654481       809
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  2729 ^2 = 7447441			     |
  2731 ^2 = 7458361 7458361-7447441= (455)24 | (455)24 +7447441 = 7458361 7458361     2731
  2741 ^2 = 7513081 7513081-7458361=(2280)24 |(2280)24 +7458361 = 7513081 7513081     2741
   		
			~~~~~~~	headings combined to save space~~~~~~~~~~~~~
  4691    ^2 = 22005481 
  4703    ^2 = 22118209 -22005481=  (4697)24 | (4697)24 +22005481 = 22118209   ^1/2   4703
  4721    ^2 = 22287841 -22118209=  (7068)24 | (7068)24 +22118209 = 22287841   ^1/2   4721
  
  5749    ^2 = 33051001   
  5779    ^2 = 33396841 -33051001= (14410)24 | (14410)24 +33051001 = 33396841   ^1/2   5779
  5783    ^2 = 33443089 -33396841=  (1927)24 |  (1927)24 +33396841 = 33443089   ^1/2   5783
  
  6841    ^2 = 46799281	   
  6857    ^2 = 47018449 -46799281=  (9132)24 |  (9132)24 +46799281 = 47018449   ^1/2   6857
  6863    ^2 = 47100769 -47018449=  (3430)24 |  (3430)24 +47018449 = 47100769   ^1/2   6863
  
  7901    ^2 = 62425801     
  7907    ^2 = 62520649 -62425801=  (3952)24 |  (3952)24 +62425801 = 62520649   ^1/2   7907
  7919    ^2 = 62710561 -62520649=  (7913)24 |  (7913)24 +62520649 = 62710561   ^1/2   7919
 
  8231    ^2 = 67749361    
  8233    ^2 = 67782289 -67749361=  (1372)24 |  (1372)24 +67749361 = 67782289   ^1/2   8233
  8237    ^2 = 67848169 -67782289=  (2745)24 |  (2745)24 +67782289 = 67848169   ^1/2   8237
    
  9719    ^2 = 94458961    
  9721    ^2 = 94497841 -94458961=  (1620)24 |  (1620)24 +94458961 = 94497841   ^1/2   9721
  9733    ^2 = 94731289 -94497841=  (9727)24 |  (9727)24 +94497841 = 94731289   ^1/2   9733
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  
 10141 ^2 = 102839881  
 10151 ^2 = 103042801 -102839881=  (8455)24 |  (8455)24 +102839881 =103042801   ^1/2  10151
 10159 ^2 = 103205281 -103042801=  (6770)24 |  (6770)24 +103042801 =103205281   ^1/2  10159
 
 11003 ^2 = 121066009  
 11027 ^2 = 121594729-121066009= (22030)24 | (22030)24 +121066009 =121594729    ^1/2  11027
 11047 ^2 = 122036209-121594729= (18395)24 | (18395)24 +121594729 =122036209    ^1/2  11047

 12941 ^2 = 167469481  
 12953 ^2 = 167780209-167469481= (12947)24 | (12947)24 +167469481 =167780209    ^1/2  12953
 12959 ^2 = 167935681-167780209=  (6478)24 |  (6478)24 +167780209 =167935681    ^1/2  12959

 13033 ^2 = 169859089 
 13037 ^2 = 169963369-169859089=  (4345)24 |  (4345)24 +169859089 =169963369    ^1/2  13037
 13043 ^2 = 170119849-169963369=  (6520)24 |  (6520)24 +169963369 =170119849    ^1/2  13043
 
 14407 ^2 = 207561649  
 14411 ^2 = 207676921-207561649=  (4803)24 |  (4803)24 +207561649 =207676921    ^1/2  14411
 14419 ^2 = 207907561-207676921=  (9610)24 |  (9610)24 +207676921 =207907561    ^1/2  14419
  
 15937 ^2 = 253987969  
 15959 ^2 = 254689681-253987969= (29238)24 | (29238)24 +253987969 =254689681    ^1/2  15959
 15971 ^2 = 255072841-254689681= (15965)24 | (15965)24 +254689681 =255072841    ^1/2  15971
 
 17851 ^2 = 318658201  
 17863 ^2 = 319086769-318658201= (17857)24 | (17857)24 +318658201 =319086769    ^1/2  17863
 17881 ^2 = 319730161-319086769= (26808)24 | (26808)24 +319086769 =319730161    ^1/2  17881
 
 19543 ^2 = 381928849  
 19553 ^2 = 382319809-381928849= (16290)24 | (16290)24 +381928849 =382319809    ^1/2  19553
 19559 ^2 = 382554481-382319809=  (9778)24 |  (9778)24 +382319809 =382554481    ^1/2  19559
 
 27739 ^2 = 769452121  
 27743 ^2 = 769674049-769452121=  (9247)24 |  (9247)24 +769452121 =769674049    ^1/2  27743
 27749 ^2 = 770007001-769674049= (13873)24 | (13873)24 +769674049 =770007001    ^1/2  27749
  
 37097 ^2 = 1376187409  
 37117 ^2 = 1377671689-1376187409=(61845)24 | (61845)24 +1376187409 =1377671689 ^1/2  37117
 37123 ^2 = 1378117129-1377671689=(18560)24 | (18560)24 +1377671689 =1378117129 ^1/2  37123

 46687 ^2 = 2179675969  
 46691 ^2 = 2180049481-2179675969= (15563)24 | (15563)24 +2179675969 =2180049481 ^1/2 46691
 46703 ^2 = 2181170209-2180049481= (46697)24 | (46697)24 +2180049481 =2181170209 ^1/2 46703
 
 50123 ^2 = 2512315129  
 50129 ^2 = 2512916641-2512315129= (25063)24 | (25063)24 +2512315129 =2512916641 ^1/2 50129
 50131 ^2 = 2513117161-2512916641=  (8355)24 |  (8355)24 +2512916641 =2513117161 ^1/2 50131
  
 60773 ^2 = 3693357529  
 60779 ^2 = 3694086841-3693357529= (30388)24 | (30388)24 +3693357529 =3694086841 ^1/2 60779
 60793 ^2 = 3695788849-3694086841= (70917)24 | (70917)24 +3694086841 =3695788849 ^1/2 60793
 
 79867 ^2 = 6378737689  
 79873 ^2 = 6379696129-6378737689= (39935)24 | (39935)24 +6378737689 =6379696129 ^1/2 79873
 79889 ^2 = 6382252321-6379696129=(106508)24 |(106508)24 +6379696129 =6382252321 ^1/2 79889
  
 82963 ^2 = 6882859369  
 82981 ^2 = 6885846361-6882859369=(124458)24 |(124458)24 +6882859369 =6885846361 ^1/2 82981
 82997 ^2 = 6888502009-6885846361=(110652)24 |(110652)24 +6885846361 =6888502009 ^1/2 82997
  
 92461 ^2 = 8549036521  
 92467 ^2 = 8550146089-8549036521= (46232)24 | (46232)24 + 8549036521 =8550146089 ^1/2 92467
 92479 ^2 = 8552365441-8550146089= (92473)24 | (92473)24 + 8550146089 =8552365441 ^1/2 92479
 
 99989 ^2= 9997800121  
 99991 ^2= 9998200081-9997800121= (16665)24 | (16665)24 +9997800121 =9998200081  ^1/2  99991
100003 ^2=10000600009-9998200081= (99997)24 | (99997)24 +9998200081 =10000600009 ^1/2 100003
100019 ^2=10003800361-10000600009=(133348)24|(133348)24 +10000600009=10003800361 ^1/2 100019

104309 ^2=10880367481 
104311 ^2=10880784721-10880367481=(17385)24 | (17385)24 +10880367481=10880784721 ^1/2 104311
104323 ^2=10883288329-10880784721=(104317)24|(104317)24 +10880784721=10883288329 ^1/2 104323

104717 ^2=10965650089 
104723 ^2=10966906729-10965650089=(52360)24 | (52360)24 +10965650089=10966906729 ^1/2 104723
104729 ^2=10968163441-10966906729=(52363)24 | (52363)24 +10966906729=10968163441 ^1/2 104729

                                           ~~~                          

Commentary

Take the squares of any prime number 5 or more, and the difference between it and any other squared prime number is always an exact multiple of 24, n24. Twenty-four, or a multiple of 24, when added to the square of any prime number, and the square root of that sum taken, will always equal the next prime number to follow. This follows from the smallest primes ( P5 or higher) to predicting and calculating the next largest known prime. Please donate your prize money to help fight world hunger, disease and the deterioration of the planet and/or for the education of that under-privileged child. Fig.2.

Postulate 3: "Butterfly Primes"

a. An overlap of all multiples of 24 (n24) on the Multiplication Matrix (Table) reveals a visual and mathematically logical pattern ("Butterfly"). The squared primes, five and up, are always positioned exactly between two symmetrically placed n24s along side the diagonal "squares" line...effectively selecting out all prime candidates such that, taking the square root of, all even numbers along one axis times the same numbers increased by 2 on the other axis (whose product must equal n24)...and adding one to that product, (Figs 3a, b and c, Table III) generates the candidates, as:

P2candidate = [Neven x (N+2)] + 1      (11)

Pcandidate = ([Neven x (N+2)] + 1)½    (12)

b. Applying the principles of Postulates 1 and 2 above eliminates all the spurious entries leaving only the squared primes.

Fig 3a:Postulate 3: 'Butterfly Primes

Fig. 3a
(click to enlarge image)
Best viewed 800x600, F-11 key

Fig 3a:Postulate 3: 'Butterfly Primes

Fig. 3b
(click to enlarge image)
Best viewed 800x600, F-11 key

Fig 3a:Postulate 3: 'Butterfly Primes

Fig. 3c
(click to enlarge image)
Best viewed 800x600, F-11 key

Table III

      
     Table III Postulate 3 


  [ Product must equal n24]                                   
   [Neven      x      (N+2)]  +  1   =  P2candidate   (P2candidate)1/2  =  Pcandidate
  ________________________________    ____________ ______________    __________
      - 
      -                                                             >=non-prime
      -
      4			  6	1	   25		  25  		  5
      6			  8	1	   49		  49		  7
     10			 12	1	  121		 121		 11
     12			 14	1	  169		 169		 13
     16			 18	1	  289		 289		 17
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
     18			 20	1	  361		 361		 19
     22			 24	1	  529		 529		 23
     24			 26	1	  625		 625	        >25
     28			 30	1	  841		 841		 29
     30			 32	1	  961		 961		 31
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~  
     34			 36	1	 1225		 1225	        >35
     36			 38	1	 1369		 1369		 37
     40			 42	1	 1681		 1681		 41
     42			 44	1	 1849		 1849		 43
     46			 48	1	 2209		 2209		 47
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
     48			 50	1	 2401		 2401	        >49
     52			 54	1	 2809		 2809		 53
     54			 56	1	 3035		 3035	        >55
     58			 60	1	 3481		 3481		 59
     60			 62	1	 3721		 3721		 61
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
     64			 66	1	 4225		 4225		 65
     66			 68	1	 4489		 4489		 67
     70			 72	1	 5041		 5041		 71
     72			 74	1	 5329		 5329		 73
     76			 78	1	 5929		 5929	        >77
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
     78			 80	1	 6241		 6241		 79
     82			 84	1	 6889		 6889		 83
     84			 86	1	 7225		 7225	        >85
     88			 90	1	 7921		 7921		 89
     90			 92	1	 8281		 8281	        >91
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
     94			 96	1	 9025		 9025	        >95
     96			 98	1	 9409		 9409		 97
    100			102	1	10201		10201	     	101
    102			104	1	10609		10609		103
    106			108	1	11449		11449		107
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
    108			110	1	11881		11881		109
    112			114	1	12769		12769		113
    114			116	1	13225		13225	       >115
    118			120	1	14161		14161	       >119
    120 		122	1	14641		14641	       >121
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
    124			126	1	15625		15625	       >125
    126			128	1	16129		16129		127
    130			132	1	17161		17161		131
    132			134	1	17689		17689	       >133
    136			138	1	18769		18769		137
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
       ~~~~~~~~~~~~~~random samples of know primes~~~~~~~~~~~~~~~~~~~~~
    786			 788	1	619369		619369		 787
    796			 798	1	635209		635209		 797
    808			 810	1	654481		654481		 809
    810			 812	1	657721		657721		 811
 
   2728			2730	1	7447441		7447441		2729
   2730			2732	1	7458361		7458361		2731
   2740			2742	1	7513081		7513081		2741
   
   4690			4692	1	22005481	22005481	4691
   4702			4704	1	22118209	22118209	4703
   4720			4722	1	22287841	22287841	4721
 
   5748			5750	1	33051001	33051001	5749
   5778			5780	1	33396841	33396841	5779
   5782			5784	1	33443089	33443089	5783
 
   6840			6842	1	46799281	46799281	6841
   6856			6858	1	47018449	47018449	6857
   6862			6864	1	47100769	47100769	6863
 
   7900			7902	1	62425801	62425801	7901
   7906			7908	1	62520649	62520649	7907
   7918			7920	1	62710561	62710561	7919

 

Commentary

Within the Multiplication Matrix (Table) there are any number of patterns found by delineating the occurrences of a particular number multiple. Here, the concern is with the number 24.

A basic, modular repeating visual pattern is found by highlighting all instances of n24 within the matrix numbers formed from the products of the 1-24 x- and y-axis numbers. This pattern is mirrored in the other three quadrants of an expanded 48 x 48 matrix, and so on...very much a fractal pattern encompassing the products of all numbers.

The original 24 x 24 fractal base pattern is itself composed of four quadrants formed by the crossings of n24s at the product lines of the 12 x- and y-axis. Because the squares of the primes all fall symmetrically on the diagonal drawn from the origin, the focus will be solely on those two quadrants hereafter...though it is to be remembered that the n24 pattern is mirrored in all quadrants as a pattern of eight internal n24s highlights. What remains in this simplification is the "Butterfly Primes" pattern...two diagonally symmetrical quadrant patterns whose eight internal, and 10 axial, n24s highlights radiate out from the center (12 x 12 =144 in this module). This pattern extends diagonally to infinity. In between each and every n24 pair which lies along side the strict diagonal "squares" line (always off the diagonal number by 1), there exists a squared prime number candidate. The pattern is of n24s shepherding the squared primes along the diagonal processional. This pattern defines and predicts all primes when the spurious candidates are eliminated by applying the principles of Postulates 1 and 2.

Twin primes ( P, P+2 ), Prime Triples ( P, P+2, P+6 or P, P+4, P+6 ) and Prime Quadruples ( P, P+2, P+6, P+8 or P, P+2, P+4, P+8 ) and other prime patterns can be seen to result from the logical beauty and order of the "Butterfly" pattern. Fig.3.

Conjecture: "Proof of Riemann's Hypothesis"

Taken together, these three postulates prove the order within the prime numbers necessary to establish a non-variance factor (n24)...the landscape at sea level (Riemann's zero points of the zeta function)...the critical line...and it is not "noisy", random or fluctuating, but constant, predictable and straight...and thus proves Riemann's Hypothesis. Each and every prime number square is shepherded along the "squares" diagonal by a series of constant, ordered and symmetrical n24s confirming the larger order of the primes within this "Butterfly" of n24s.

New Prime Definition = new "Prime Number Theorem"

A prime number is any natural whole number, greater than one, which is reducible only by itself and one, and, is always separated from the square of all other primes (except P2,3,5 ) by multiplies of 24. The infinity of primes can not exceed the infinity of n24.

Conjecture: Primes and the Inverse Square Law

The Inverse Square Law, 1/r2, so fundamental to our physical world...witness gravity and electromagnetism, light and sound, energy, etc...is built on the "odd number summation series", that is, as r, the radius (or distance) increases as 1,2,3,4,5,... the sequential difference between the squares, as r2, follows as 3,5,7,9,11,.... As the distance, r, is doubled, the influence diminishes by 1/4. When r is tripled the influence is reduced by 1/9, when r is quadrupled, the influence drops to 1/16 of the original, and so on. The odd numbers: 1...3...5...7...9... and so on, when added to each other form the sums: 1...4...9...16...25... and so on, forming the "odd number summation series"...the square roots of which form the series: 1,2,3,4,5 and so on.

The "odd number summation series can easily be seen as the diagonal "squares" line on any of the above figures. The odd numbers which form it (when squared), are made of prime and non-prime numbers. The non-prime numbers can be shown to be made of the product of two or more prime numbers. Table IV.

It follows that all odd natural whole numbers, greater than one, are either prime or composed of the product of two or more prime numbers. See the (1) Caldwell and (4) Brooks reference links below.

Table IV

 
    Table IV. Primes and the Inverse Square Law     

     All odd          (Nodd)2      Non-Prime             
     numbers,         generates    Nodd(x-axis)
     Nodd(x-axis)        the odd      numbers are
     3-100...,        Inverse      the product
     form the         Square       of two or more
     difference	      Numbers      Prime numbers, P
     between the                  
     Inverse                        
     Square
     Number's
     "odd number
      summation
      series"
     ______           _______   __________________ 

       
      3    squared =     9    
      5    squared =    25     
      7    squared =    49   
      9    squared =    81        9 = 3x3
     11    squared =   121   
     13    squared =   169 
     15    squared =   225       15 = 3x5
     17    squared =   289    
     19    squared =   361    
     21    squared =   441       21 = 3x7
     23    squared =   529     
     25    squared =   625       25 = 5x5
     27    squared =   729       27 = 3x3x3
     29    squared =   841
     31    squared =   961
     33    squared =  1089       33 = 3x11
     35    squared =  1225       35 = 5x7
     37    squared =  1369    
     39    squared =  1521       39 = 3x13
     41    squared =  1681     
     43    squared =  1849  
     45    squared =  2025       45 = 3x3x5
     47    squared =  2209
     49    squared =  2401       49 = 7x7
     51    squared =  2601       51 = 3x17              
     53    squared =  2809   
     55    squared =  3025       55 = 5x11
     57    squared =  3249       57 = 3x19
     59    squared =  3481      
     61    squared =  3721     
     63    squared =  3969       63 = 3x3x7
     65    squared =  4225       65 = 5x13
     67    squared =  4489     
     69    squared =  4761       69 = 3x23
     71    squared =  5041 
     73    squared =  5329    
     75    squared =  5625       75 = 3x5x5
     77    squared =  5929       77 = 7x11
     79    squared =  6241 
     81    squared =  6561       81 = 3x3x3x3
     83    squared =  6889     
     85    squared =  7225       85 = 5x17
     87    squared =  7569       87 = 3x29
     89    squared =  7921     
     91    squared =  8281       91 = 7x13
     93    squared =  8649       93 = 3x31
     95    squared =  9025       95 = 5x19
     97    squared =  9409    
     99    squared =  9801       99 = 3x3x11
     and so on

Conclusion

By using an expanded Multiplication Matrix (Table) and plotting the patterns of the prime numbers on the axis and their squares in the matrix, a strict order based on a factor of 24 was found which is both numerically and visually logical. The difference in squared primes of multiples of 24 allows for the validation of all primes past and the prediction and calculation of all new primes to be easily accomplished. Three Postulates and a new Prime Number Theorem have been offered to summarize and codify these findings...and taken together they define a beautiful ordered pattern within the primes revealing the proof of Riemann's Hypothesis.

References

The three excellent website references below provide outstanding information, presentation and resources about primes.

1. Caldwell, Chris, The Prime Pages,
http://primes.utm.edu

First and last source to check out everything you ever wanted to know about primes-history, glossary, proofs, types, lists, resources and more. Well referenced and up to date.

2. du Sautoy, Marcus, The Music of the Primes,
http://www.musicoftheprimes.com

A beautiful site which educates you as you go along the melody of mathematical thought. Particularly insightful presentation of imaginary numbers and the musical landscape metaphor elucidating Riemann's pursuit of the great ordered pattern of the primes.

3. Watkins, Matthew R., Number Theory and Physics,
http://www.maths.ex.ac.uk/~mwatkins/zeta/physics.htm

Embedded in the bigger picture of number theory and its relationship to physics, this wonderful site both teaches and inspires by relating the history of numerical and physical thought by their authors to a contemporary presentation of those ideas. Full of resources and great quotes.

Additional writings on art, math and physics by the author can be found at:

4. Brooks, Reginald, Art Theory 101,
http://www.brooksdesign-cg.com/Code/Html/arthry2.com


Additional and valuable links added post-publication:

5. Alfeld, Peter, http://www.math.utah.edu/~alfeld/

6. Chamness, Mark, http://alumnus.caltech.edu/~chamness/Prime.html

7. Edgington, Will, http://www.garlic.com/~wedgingt/mersenne.html

8. Heinz, Harvey, http://www.geocities.com/~harveyh/primes.htm

9. Leatherland, Adrian J.F., http://yoyo.cc.monash.edu.au/~bunyip/primes/

10. The Mathematical Association of America, http://www.maa.org

11. O'Connor, John and Edmund Robertson, http://www-history.mcs.st-and.ac.uk/history/HistTopics/Prime_numbers.html

12. Peterson, Ivars, http://www.sciencenews.org

13. Woltman, George, http://www.mersenne.org/prime.htm (GIMPS)

_________________________________________________________________________________________

Copyright 2005, Reginald Brooks. All rights reserved.

Use

This paper and all its contents © 2005, Reginald Brooks. All rights reserved. This work and all its content are, to the best knowledge of its author, original and solely of and by the author at the time of its creation. Permission is hereby granted for single copies to be made for personal, non-commercial use for students and teachers of schools, colleges and universities provided that: either the entire paper, including figures and tables, is kept intact; or, any extracts of the text, or figures or tables (in part or whole), be properly and visibly cited as to authorship and source.

Special thanks to Calc98 (http://www.calculator.org/), whose software calculator eloquently handled the big digits after running out of space on my trusty hand-held Texas Instruments Slimline TI-35 Solar Scientific Calculator

Go to Butterfly Prime Directive (white paper) or
Butterfly Prime Determinant Number Array(white paper)
Butterfly Primes ~Prejudicial Numbers~ (new media net.art)




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