2.4   Equation Categorization and Inter-Conversion.

With the quadratics of the previous two Sections and other similar equations in the literature, (plus additionally those developed via preparation of this article), it is now possible to categorize these equations and show how they can be inter-converted. This will help in the search for more.

2.4.1  Categorization.

Type (i) - Full Prime Generators. - These are equations that generate prime numbers exactly according to the criteria set in Section 2.1. An example is Euler's equation, (2.5).

Type (ii) - Sub Prime Generators. - These are equations that generate prime numbers for some initial part of the criteria concerning the variable n but then fail. An example is

N = 2n2 + 4n + 31
(2.12)
This generates prime numbers for n = 0 to 3 and then fails.

Type (iii) - Symmetrical V Prime Generators. - These are equations that generate prime numbers in descending order for n = 0 to (a0/2 - 1), and then the same prime numbers in ascending order for n = a0/2 to (a0 - 2). They do not meet the criteria of Section 2.1. An example is

N = n2 - 79n + 1601
(2.13)

Type (iv) - Non-Symmetrical V Prime Generators. - These are equations that generate prime numbers in descending order for n = 0 to ni, and then a greater or lesser number of prime numbers in ascending order from n=ni+1 to ni+m. They do not conform to the criteria.

2.4.2  Inter-Conversion of Equation Categories.

From Type (ii), Sub Prime to Type (i), Full Prime. - In some cases a Sub Prime equation will generate a sub-set of a Full Prime equation. The following process will test any potential Sub Prime equation for this and make the conversion. Consider a generalised Sub Prime equation thus

N = a2 n2 + a1 n + a0
(2.14)

If the converted Full Prime equation is obtained by putting n = n - b, where b is some integer to be determined, then

N = a2(n-b )2 + a1(n-b ) + a0
(2.15)
which expands to
N = a2n2 - ( 2a2 b - a1) n + a2b2 - a1b + a0
(2.16)

If (2.16) is a Full Prime Generator, then, in accordance with the criteria, the constant term must be a prime number P, so that

a2b2 - a1b + a0 - P = 0
(2.17)
Solving for b
b = a1 ± { a12 - 4a2 ( a0 - P ) }1/2
2a2
(2.18)

The test/conversion process is now completed by finding the lowest prime value for P that makes b an integer. This process is simply one of moving the N axis along the n axis to the lowest prime on the curve to maximise the number of primes generated. However, it should be noted that even though a value for P can be found, the resulting polynomial may not be Full Prime.

Example.

Consider the Sub-Prime equation

N = 2n2 + 48n + 317
(2.19)

which generates primes starting with 317 for n = 0, but does not meet the criteria. With a2 = 2, a1 = 48 and a0 = 317 substituted in (2.18) there results


b =
48 ± 
Ö
 
  -232 + 8P
 
4
 (2.20)
The smallest prime value of P to make b an integer is 29, with which (2.20) becomes
b = 12
(2.21)
Substituting this into (2.16) finally gives
N = 2n2 + 29
(2.22)
which is Legendre's equation and is Full Prime.

From Types (iii) and (iv), V prime to Full prime. - V Prime equations can also be converted to Full Prime via an identical process to that above with the minor difference that the catalyst term is n + b. In that case the equivalent of (2.16) is

N = a2n2 + (2a2b - a1) n + a2b2 - a1b + a0
(2.23)

Note that the constant term in (2.23) is identical to that in (2.16) so that the test/conversion equation of (2.18) applies in this case also.

Example.

Consider the Symmetrical V Prime Generator

N = n2 - 79n + 1601
(2.24)
With a2 = 1, a1 = 79 and a0 = 1601 substituted in (2.18) there results


b =
79 ± 
Ö
 
 -163 + 4P
 
2
(2.25)
With P = 41 this becomes
b = 40    or    39
(2.26)
Substituting b = 40 into (2.23) gives
N = n2 + n + 41
(2.27)

which is Euler's equation and is Full Prime. Note that substituting b = 39 into (2.23) gives


N = n2 - n + 41
(2.28)

which is Legendre's version of Euler's equation and is Non-Symmetrical V Prime. Applying the conversion process to (2.28) gives

b = 1    or    0
(2.29)

With b = 1, (2.28) is converted to (2.27) and with b = 0, (2.28) sensibly does not change.

The categorisation and conversion process should be of some assistance in the challenge of the next section.



M1 Version 1.0.0
Ó P.G.Bass, March 2009
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