PRIME NUMBERS -

GENERATION BY SINGLE

VARIABLE POLYNOMIAL EQUATIONS


Peter G.Bass



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

ABSTRACT

The purpose of this paper is to identify, under a strict set of control conditions, those single variable polynomial equations of order 2 or higher, that generate only prime numbers.



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

1.0  Introduction.

Since their discovery, prime numbers have been the subject of intensive study by both professional and amateur mathematicians alike. The nature of these studies has ranged from sequence characteristics through general distribution within the complete set of integer numbers, through to their generation by a variety of equation types.

With regard to the latter, Legendre proved that there is no rational algebraic function in a variable n which would generate all the primes for all n. Similarly, Goldbach proved in 1752 that no polynomial with integer coefficients can generate even a subset of primes for all n. Consequently, study in this area has concentrated on finding those equations which give a sub-set of all primes for some n.

The primary purpose of this paper is to determine, under a specific set of control criteria, those single variable polynomial equations of order 2 or higher, that will generate only prime numbers for a range of integer values of the single variable n from 0 upwards.

The initial secondary objective is to find those equations which as a group combine to generate all the 26 prime numbers from 0 to 100. This is followed by an invitation to readers to participate in finding additional equations to generate all the prime numbers from 101 to 200.

2.0  Prime Number Generation.

2.1  Preamble and Establishment of Control Conditions.

The first matter is consideration of the number zero. Zero appears in all branches of science and as such has long been accepted as a member of the complete set of numbers. Insofar as it lies equi-distant between -1 and +1 determines that it must also be an integer. The remaining question is whether it is prime.

According to the formal definition of a prime number, viz. any number that can only be divided by itself and unity, zero would not classify as prime for the following reasons.

Albeit that


0
0
(2.1)
is indeterminate, taking

0
1
 = 0
(2.2)
any other number divided into zero gives

0
n
 = 0
(2.3)

the same value as (2.2) and therefore, apart from (2.1), zero can be divided by any and all numbers, rational and irrational, to give an integer value. Zero is the only number that possesses this characteristic. Accordingly, zero is not prime but, must be included as an integer value, the first, in the single variable n in the polynomials to be determined.

The second matter to be discussed is the control criteria under which these polynomials are to be obtained. Representing these equations thus,


N = am nm + a( m-1 ) n( m-1 ) + ¼ + a0
(2.4)

The criteria to be set against suitable variants of (2.3) are threefold as follows

(i) The coefficients ai must all be real and integer.

(ii) Clearly, when in (2.4) n = a0, N must be composite. The criterion to be applied to n is therefore that all equations must generate only prime numbers for all values of n from zero up to at least (a0 - 2), (this ensures that all equations will generate at least (a0 - 1) prime numbers). In addition, all equations must generate at least four prime numbers.

(iii) When n = 0, N = a0 so that a0 itself must be a prime number.

2.2  A Brief Resume of Existing Equations.

A great many equations have been found that generate a small range of prime numbers, but only a few conform to the criteria above. The most famous of these is that found by Euler in 1772, viz.


N = n2 + n + 41
(2.5)

Equation (2.5) generates only prime numbers for n = 0 to 39. There are two variants of (2.5) that conform to the criteria, the first is


N = n2 + n + 11
(2.6)

which generates prime numbers for n = 0 to 9. The second variant is


N = n2 + n + 17
(2.7)

This equation generates prime numbers for n = 0 to 15.

A further equation discovered by Legendre is


N = 2n2 + 29
(2.8)

which generates prime numbers for n = 0 to 28, and a variant is


N = 2n2 + 11
(2.9)
This generates prime numbers for n = 0 to 10.

The prime numbers generated by the above five equations, (2.5) to (2.9), are shown in the table below. Note that the order of listing is 1st by a0, 2nd by a1, and 3rd by a2.


EquationPrime Numbers Generated
UniqueRepeats
N=2n2+1111, 13, 19, 29, 43, 61, 83, 109, 139, 173, 211 
N=n2+n+1117, 23, 31, 41, 53, 67, 10111, 13, 83
N=n2+n+1737, 47, 59, 73, 89, 107, 127, 149, 199, 227, 25717, 19, 23, 29, 173
N=2n2+2979, 157, 191, 229, 271, 317, 367, 421, 479, 541, 607, 667, 751, 829, 911, 997, 1087, 1181, 1279, 1381, 1487, 159729, 31, 47, 61, 101, 127
N=n2+n+4141, 71, 97, 113, 131, 151, 197, 223, 251, 281, 313, 347, 383, 461, 503, 547, 593, 641, 691, 743, 797, 853, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 160143, 47, 53, 61, 83, 173, 421, 911

Table 2.1 - Prime Numbers Generated By Equations (2.5) to (2.9).


It should be noted here that as more equations are identified and inserted into the Table according to the listing order above, the distribution between Ünique" and "Repeats" in this Table will change. Some equations may thereby become redundant and removed to be added to the redundant list in Appendix A.

2.3  Equations to Complete the Set Required to Generate the First 26 Prime Numbers Between 0 and 100.

Reference to Table 2.1 shows that there are only five numbers not covered by the equations listed. They are 1, 2, 3, 5, and 7, i.e. the first five. Initially only quadratic equations are being considered so that it is only 1, 3, 5, and 7 that need to be considered here. This is because the prime number 2 cannot be generated by a quadratic, except when n = 0, but which thereafter does not conform to the criteria set.

There are two equations required to generate the above four primes, the first is


N = n2 + n + 1
(2.10)
which is clearly a simple variant of Euler's equation, (2.5). The second equation is

N = n2 + 3n + 1
(2.11)

Eq, (2.10) generates 1, 3 and 7, (plus repeat 13), while (2.11) generates 5, (plus repeats 1, 11, 19, 29 and 41). Thus together with (2.5) to (2.9), (2.10) and (2.11) complete the generation of all the 26 prime numbers between 0 and 100.


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

On to the Next Section:- Categorization

Back to the Introduction to this Paper:- Prime Numbers

Back to the Home Page for this Site:- Home