S9. 2-D / 3-D / 4-D Hyperbola's

An Approach to Prime Numbers , the L.C.M. , and the Zeta Functions

by Frank C. Fung - 1st published in May, 2009.

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Section IX - Singularities and Symmetries in { 2-D / 3-D / 4-D Hyperbola's }

Summary for the Section :

In this Section , we shall look the symmetric properties of hyperbola ,
- to get a better feel on using the hyperbola's to estimate { prime number frequencies } .
We shall also touch-on briefly ideas on { prime numbers } brought forth by Carl F. Gauss .

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The 2-D Hyperbola :

Let us now look at the 2-D Hyperbola's , concentrating on the { nodal points } here :
- where the number [ N ] is the product of 2 { prime numbers } .
For the [ square-of-primes ] , e.g. [ 4 ] , [ 9 ] , [ 25 ] , [ 49 ] , etc. :
- we can use the format :
  - [ N ] = A*A ,
    
    and these are singular in nature .
For the [ non-squares ] , e.g. [ 6 ] , [ 10 ] , [ 14 ] , [ 15 ] , etc. :
- we can use the format :
  - [ N ] = B*C = C*B ,
    
    and these are plural / bi-symmetric in nature .

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The 3-D Hyperbola :

Let us now look at the 3-D Hyperbola's , concentrating on the { nodal points } here :
- where the number [ N ] is the product of 3 { prime numbers } .
For the [ cube-of-primes ] , e.g. [ 8 ] , [ 27 ] , [ 125 ] , [ 343 ] , etc. :
- we can use the format :
  - [ N ] = A*A*A ,
    
    and these are singular in nature .
For the [ square-on-single ]'s , e.g. [ 12 ] , [ 18 ] , [ 20 ] , [ 28 ] , etc. :
- we can use the format :
  - [ N ] = B*B*C = B*C*B = C*B*B ,
    
    and these are tri-symmetric in nature .
For the other [ general numbers ] , e.g. [ 30 ] , [ 42 ] , [ 70 ] , [ 105 ] , etc. :
- we can use the format :
  - [ N ] = D*E*F = E*F*D = F*D*E = E*D*F = D*F*E = F*E*D ,
    
    and these are hexa-symmetric in nature .

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The 4-D Hyperbola :

Let us now look at the 4-D Hyperbola's , concentrating on the { nodal points } here :
- where the number [ N ] is the product of 4 { prime numbers } .
For the [ 4th-power-of-primes ] , e.g. [ 16 ] , [ 81 ] , [ 625 ] , [ 2401 ] , etc. :
- we can use the format :
  - [ N ] = A*A*A*A ,
    
    and these are singular in nature .
For the [ Cube-and-Single ] , e.g. [ 24 ] , [ 40 ] , [ 54 ] , [ 56 ] , etc. :
- we can use the format :
  - [ N ] = B*B*B*C = B*B*C*B = B*C*B*B = C*B*B*B ,
    
    and these are quad-symmetric in nature .
For the [ Square-and-Square ] , e.g. [ 36 ] , [ 100 ] , [ 196 ] , [ 225 ] , etc. :
- we can use the format :
  - [ N ] = D*D*E*E = D*E*E*D = E*E*D*D = E*D*D*E = D*E*D*E = E*D*E*D ,
    
    and these are hexa-symmetric in nature .
For the [ Square-and-2-singles ] , e.g. [ 60 ] , [ 84 ] , [ 90 ] , [ 126 ] , etc. :
- we can use the format :
  - [ N ] = F*F*G*H = F*G*H*F = G*H*F*F = H*F*F*G
    = F*F*H*G = F*H*G*F = H*G*F*F = G*F*F*H
    
    = F*G*F*H = F*H*F*G = H*F*G*F = G*F*G*H
    
    and these are 12-symmetric in nature .
For the other [ general numbers ] , e.g. [ 210 ] , [ 330 ] , [ 390 ] , [ 770 ] , etc. :
- we can use the format :
  - [ N ] = J*K*L*M = K*L*M*J = L*M*J*K = M*J*K*L
    = J*K*M*L = K*M*L*J = M*L*J*K = L*J*K*M
    
    = J*M*K*L = M*K*L*J = K*L*J*M = L*J*M*K
    
    = J*M*L*K = M*L*K*J = L*K*J*M = K*J*M*L
    
    = J*L*K*M = L*K*M*J = K*M*J*L = M*J*L*K
    
    = J*L*M*K = L*M*K*J = M*K*J*L = K*J*L*M
    
    and these are 24-symmetric in nature .

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The use of 2-D / 3-D / 4-D Hyperbola's in { Distribution of Prime Numbers } :

In our Fast-Track procedure to identify { prime numbers } in Section VII ,
- we've used [ 2-D / 3-D hyperbola's ] as { boundary / dividing lines } in 2-D / 3-D { prime multiplication tables / grids } ,
  - for the purpose of :
    - Bringing-in { non-prime numbers } back into the [ Base Candidate Row ] in the { candidate proposing stage } ; and
    - Knocking-off { non-prime numbers } during the { consolidating stage } .
Clearly , 4-D and higher-ordered hyperbola's will be involved as we move-up-the-ladder on the use of the { product-of-primes modulo's } .
Since the { area under the curve } for the [ 2-D hyperbola ] is related to { logarithms } ,
- { logarithms } is logically the first-place-to-look-at when doing the { distribution of prime numbers } .
But a word of caution here :
- when using { 2-D hyperbola's } , we would be comparing [ areas ] here ;
- when using { 3-D hyperbola's } , we would be comparing [ volumes ] here ;
- when using { 4-D hyperbola's } , we would be comparing [ 4-D hyper-volumes ] here ;
  
  And higher-order hyperbola's will involve [ multi-dimensional hyper-volumes ] .
We need to get-a-fix on the proper procedure before moving-in on the said-problem at-hand .
But even after we've got the proper handle on the problem here , we still need to resolve :
- the issue of the skewness on the { distribution of prime numbers } within the { finite range at-hand }
- the issue of singularity vs. multiplex-symmetry in the [ prime-product nodal points ] :
  - in the [ 2-D plane ] , [ 3-D space ] , [ 4-D hyper-space ] , etc. ; as mentioned above .

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Tracking the work of C.F. Gauss :

Gauss intially proposed this for the { distribution of prime numbers } :
Gauss later revised his proposal with this refinement :
- where :
Both of these involve { logarithms } .
In Part V of the paper , we will be tracking partially the work of G.F.B.Riemann .

We note , at this point , that Riemann also brought-in the { power-of-primes } into the picture , via his { f(x) function } :
- and this might be an improvement , to the extent that :
  - { square-of-primes } are on the { 1, 1 } diagonal in the [ 2-D plane ] , and are singular in nature ;
  - { cube-of-primes } are on the { 1, 1 , 1 } diagonal in the [ 3-D space ] , and are singular in nature ;
  - { 4th-power-of-primes } are on the { 1, 1, 1, 1 } diagonal in the [ 4-D hyper-space ] , and are singular in nature .
May be Riemann already had a better vision on the process of knocking-off { non-prime numbers } ?
- Simply don't know and can't tell .