Prime Numbers Classification with Linear and Quadratic Forms

It is known, that any prime number has presentation in linear and quadratic forms. These properties may be used for finding class subsets of prime numbers. For that it is showed, that all prime numbers simple quadratic forms consist of 2 2 , 1, 2, 3 a mb m + = ± . On these grounds it is examination for variants of prime numbers classification. It is discovered eight non-intersecting subsets of prime numbers, in conformity with equivalence classes modulo 24. The proposed classification is used for analyses Mersenne and Fermat numbers and composite numbers.


Introduction
The fundamental theorem of arithmetic states that any positive integer can be represented in exactly one way as a product of prime numbers. The difference between primes are come to light if investigate their linear and quadratic forms. For instance, there are two subsets of primes with linear form 4 1 n + and with linear form 4 3 n + . It is practically simplest prime number classification with equivalence classes modulo 4. It is used for elementary analyzing and proving in Number Theory. But anybody can to see in literature the absence further researching on classification theme. This paper is tried to fill this gap with more detailed classification with hope to use it for more complicated problems. The offered classification is based on the old researches, begun by Albert Girard (1595-1632) and Pierre de Fermat (1601-1665) and in detail analyzed in [1, §1.7]. It was found, that only the prime numbers of line series 4 1 n + have a single presentation as a sum of two squares. Also appeared, that only prime numbers of line series 8 1 n + or 8 3 n + have single presentation as a sum

Prime Numbers with Other Quadratic Form
Without ambiguity we will accept, that prime numbers 2 2 24 The sequence of such quadratic forms is endless, but has determination by its initial decision.

Basis and Variants of Classification
The stated allows offering simple and evident classification of the prime numbers, separated on four partly intersecting sets of circular, elliptic (two sets) and hyperbolic prime numbers.
It should be noted that in this classification the linear form 8 1 n − has not an obvious presentation. It is so, because these prime numbers are disintegrated on two non-intersecting sets depending on the value of residue ( ) for prime numbers characterizing. On our basis it maybe to perfect and simplify the classification presented in Table 1, using only the non-intersecting sets of prime numbers.    Table 2 we see the possession for more detailed classification, if for basis to take non-intersecting subsets corresponding to the non-intersecting residue classes modulo 24.
We will designate quadratic forms as R -Circular, 1, 2 E E -Elliptic 1 and 2, G -Hyperbolic. It is Diophantine equations modulo 24, and similar problems are easy solved with computer programs Mathematica. We use function Reduce, for solving equations over the integers modulo 24, and there is simplification, as sense has only alternative: is decision or not.

Applications
Just as above, we may use classification for researching any subsets of primes with other linear, quadratic and polynomial forms. For instance, consider primes with above mentioned quadratic form 2  The proof is evident, as we use primes classification Table  3 with non-intersecting subsets.
The got classification we will apply for the analysis of some problems of Number Theory with composite numbers. In the beginning we will consider the set of prime numbers 2  In general case for arbitrary numbers classification can be produced by means of computers with the corresponding programs.
Author was use the program Mathematica, working with the tasks of Number Theory.
As an example we will consider a number In the beginning we determine, is that number prime or composite, and for this purpose there are standard programs of Mathematica.
If it is prime number, then further we calculate ( ) mod 24 a N ≡ , and classify it in obedience to , that for the decision of that task the computer works a few hours, and virtual storage is not enough for that. It was unexpected for the developer of program Stan Wagon, one of authors of guidance [3], and author of that article appealed to him. Wagon reported through two weeks, that on the basis of the new improvements of algorithm of «HalfGCD» [4] Daniel Lichtblau perfected program, and now it decides the task for 0, 3 secs. However to this time an author was worked out private program, on Wagon proposal afterwards named QRM (Quadratic Representations Meshkoff). It makes use only of formulae (2) and standard functions of Mathematica 1) Factor Integer and 2) Quadratic Representation. For that task QRM on the Wagon computer has time 0, 32 secs. The decision another way we can to get with Mathematica program Reduce, but the calculation time is about 5 secs, so it is not advantageous for big numbers.
Further comparisons for different numbers showed, that on the Wagon computer QRM some yields on speed. However on the private computer of author for QRM time of decision was 0,15-0,16 secs, and program of DL (sending from Wagon) did not work, as it was based on internal codes of firm Mathematica. In addition, QRM decided a task for the even values of N, and also for cases, when N is product of prime numbers and part of them is not circular. These numbers are eliminated from product, and the task was decided for product of remaining circular prime numbers.
In like manner it can be built programs for other three quadratic presentations. Then for the examined task in the beginning we find

Conclusions
It would be much more difficult to decide similar tasks, having not the classification offered higher. That classification is natural and scientific, so it is at the same time the result and the important instrument of science research. If before primes as some objects seems to be grey or black and white ( 4 1 n + and 4 3 n + ), now it is as we see the objects of eight colors. The classification for composite numbers of two primes consists of these colors eight objects and 56 objects of two colors.
Composite numbers of three and more primes so are significantly complicate in sense of many colors objects, and there is one of explanations why Number Theory problems complication fast grow up for big numbers.
Author has hope, if offered classification find application in theoretical researching on Number Theory, as above showed instances confirm it, as well as applications for optimizing calculations and computer programs in this field. This work is pioneer on that theme, so references are scarce. All well-known material on Number Theory, used in this work without references from MathWorld -A Wolfram Web Resource, especially its article: Eric W. Weisstein «Prime Number».
This work is first published on English of author's manuscript 2007 year, now revised and corrected.