Ÿ Preface of author

Number theory is one of the very prolific branches of mathematics that is rooted in the depth of human history. When human being started to count, he found out number theory. Background of theory goes back to about four thousand years ago. Babylonian and Sumerian civilization that some remain writings from that time have some signs of it. Even we can claim that previous record of number theory has deeper roots in human history. Number theory became prolific the same as human knowledge improved and mathematics became expanded. After counting times in Babylonian and India, this theory was propounded as mathematics enigmas in ancient Greece and enamored a lot of scholars and philosophers, so that it led them to worship the numbers that as an example we can mention Pythagoreans.

Now-days the start of  2lst century, Number theory (specially with the solution of Fermat’s last theorem by Andrew Wiles) is taken into account as one of the prolific mathematician and leader among mathematics branches.

Numbers star glitter in splendid mode, as so that it has attracted the entire world to it. Clear statements of Hardy (famous English mathematician) are a reason that indicates more attention to number theory: "Elementary number theory must be one of the most important subjects for primitive education in mathematics.

It does not need wide previous information. Its subject is palpable and familiar. Ratiocinative methods which it uses are simple, general and a little and it has not unique in mathematic science to excitation the natural curiosity since of human. A thoughtfully teach for one month in number theory is two times more sufficient and useful and at least ten times more entertain than the same time teaching (calculus for engineers)."

It is difficult to teach prime numbers to individuals and groups, and so its attraction has absorbed a lot of mathematicians to itself. Solving problems in different fields of number theory specially this field is a stony and dark path and it is hard to find the way even narrow one to continue the work.

Second, because apparently, acquaintance with rules of prime numbers can not help to another science and industries.

Although some mathematicians started this way and they procured approximate success.

For finding a function to generate prime numbers so that it gives only prime numbers for every natural number, a lot of works has been done that most of them are built on Wilson’s theorem.

Since this theorem is based on knowing some numbers in the format of n! , it has only value of a pure theorem. Therefore identification a number like 69! , it is necessary to know a 99 digits number, because 69! 1.7×1098.

Up to now, such explicit function has not been found and we can mention only Niven’s theorem that is extended from Mills’s theorem which proves the existence of a prime numbers generator function.

The following subjects are the most important topics that include most of the trying and researches in history of mathematics:

ŸForming the table of prime numbers and definition the least factor of composite numbers

ŸThe identification function of prime numbers and the ways of factorization the large composite numbers.

ŸThe generating function of prime numbers ŸDetermining of prime number ŸThe function of determining the number of the prime numbers less than arbitrary number "N" exactly or approximately ŸSolution of Riemann's Zeta equation ŸSearch for another Fermat’s and Mersenne’s prime numbers and factorization of these numbers ( and Mp).

ŸTrying for proof the conjectures of being infinity of the twin prime numbers

ŸTrying for proof the conjectures of Goldbach and Hardy

ŸSolution of the Diophantine equations and linear and nth degree non linear systems

ŸSolution of the equations and congruence systems

Ÿ350 years attempts for proving the Fermat’s last theorem.

This theorem proof was the major in number theory by Taylor–wiles. And the proof of this theorem by Taylor-Wiles was caused a large movement in number theory.

ŸPresenting the algorithms and easy soft wares with possible high speed for finding greatest unknown prime numbers and calculating the determinants

Ÿ It is more than 2000 years that the prime numbers and the problems about it, is one of the most important field in mathematics and it has been attended by professional and technical mathematicians. But, there are a lot of unsolved problems in this field that we will mention some of them later.

The statements of this book are presented as everyone can inform the historical process of the basic problems of number theory, and then inform from the latest researched results that have been made during 20 years of author’s investigations.

Appreciation and Gratitude

I would appreciate my wife and two children that with out their assistance and support completion of this book was impossible. Also I would sincerely express my appreciation and gratitude from honorable persons in charge of Brill/VSP publications specially Drs Doutzen Abma (Acquisition Editor of VSP), Dr Matthias W.C.Wahls (Business Development Manager of Brill), Dr Hans van der Meyden (Production Manager of VSP) and Miss Renee Otto (Publishing Assistant of VSP) for their patience and support in various phases of progress in publishing the book.