Saturday, 3 March 2012

On equations intended to generate only prime numbers

Like most people who are fascinated by mathematics and analysis, I enjoy contemplating the nature of prime numbers. Prime numbers are those that cannot be factored into rational numbers other than themselves or by one.

Throughout maths long history there have been many, many attempts at creating equations that will produce only prime numbers and nothing else. Some famous (or infamous) examples include such equations as:

P(x) = 2x + 1 who more generalized linear form is P(x) = px + q

but there others, such as

the quadratic P(x)  = x2 + 1,

which stipulates that a necessary condition that in order for P(x) to be prime is that x must end in the digits 4 or 6; (16) + 1 = 17; (36) + 1 = 37, and so on...

then there is the Mersenne function: P(x) = 2x - 1

or the Fermat function: P(x) = 2x + 1.

There are no shortages of very productive equations, including ones created by Euler.

The thing I want to say here is that basic mathematical concepts like the restriction principle and closure of number domains with respect to the fundamental operations of arithmetic do not seem to allow the generation of only prime numbers. To quote, again, Tobias Dantzig:

For instance, we could terminate the natural sequence at the physiological and psycholoogical limits of the counting process, say 1,000,000. In such an arithmetic addition and multiplication, when possible, would be associative and  commutative; but the operations would not always be possible. Such expressions as (500,000 + 500,001) or (1000 x 1001) would be meaningless, and it is obvious that the number of meaningless cases would far exceed those which have a meaning. This restriction on integers would cause a corresponding restriction on fractions; no decimal fraction could have more than 6 places, and the conversion of such a fraction as 1/3 into a decimal fraction would have no meaning. Indefinite divisibility would have no more meaning than indefinite growth, and we would reach the indivisible by dividing any object into a million equal parts. (p. 76)

The two basic conditions alone (the restriction principle and closedness of number domains) seem to preclude equations that could generate only prime numbers.

But that is not to say it is futile to contemplate the prime numbers. Far from it. These are mathematical entities which belie such "restrictions" by the simple virtue of their significance to number theory. Prime numbers are fundamental to number analysis and have far-reaching and interesting ramifications both within pure and applied mathematics.

Besides, any intelligent person can perceive the unique and fascinating qualities of such numbers. Mathematical entities invariably hold infinite possibilities for those with the perseverance and imagination to pursue them, those with the chutzpah to negotiate with "the legendary dragon guarding the entrance to the enchanted garden". (p. 64)

Jay

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