**Number theory** is a branch of mathematics that deals with the properties of integers. The application of the elementary arithmetic or number theory includes a generalization of arithmetic, the theory of Diophantine equations, the analytic number theory and algebraic number theory.

From antiquity to the seventeenth century, the theory of numbers was asserted as undergraduate discipline and came out without other mathematical sub-regions. Its only tools were the properties of the integers, in particular prime factorization (Fundamental Theorem of Arithmetic), divisibility, and calculating with congruence. Such a pure approach is also known as elementary number theory. Important results can be achieved using elementary methods, which are small Fermat’s theorem and its generalization, the set of Euler, the Chinese remainder theorem, the set of Wilson and the Euclidean algorithm.

If you write research paper on number theory, you should know that even today, research is being conducted on specific issues to divisibility, congruence, and the like with elementary number theory methods. Likewise, an attempt is evidence to number theory that use of further methods in elementary terms to “translate” from which new insights can emerge. An example is the elementary observation of the number-theoretic functions as the MÃ¶bius function and the Euler Phi function.

The first written evidence of number theory dates back up to about 2000 BC. The Babylonians and Egyptians knew at that time already the numbers less than one million, the square numbers, and some Pythagorean triples.

However, the systematic development of number theory began only in the first millennium BC in ancient Greece. Most Outstanding Representative is Euclid (about 300 BC), who invented method of mathematical proof introduced in the theory of numbers. His most famous work, The Elements of Euclid, was used up in the eighteenth century as a standard textbook for geometry and number theory. Volumes 7, 8 and 9 deal with this number-theoretic issues, including the definition of a prime number, a method for calculating the greatest common divisor (Euclidean algorithm) and the proof of the existence of infinitely many prime numbers (set of Euclid).

In the year 250 BC, the Greek mathematician Diophantus of Alexandria occupied himself first with the equations named after him, he was trying to cut down known cases of linear substitutions. So that he could actually solve some simple equations. Diophantus’ major work is the Arithmetica.

The Greeks threw on many important arithmetical questions, which to this day are still in part unresolved (such as the problem of twin primes and that of perfect numbers) or their solution took many thousands of years to complete and which are examples of the development of number theory.

With the downfall of the Greek states, the heyday of number theory in Europe also went out. From this time only the name of Leonardo di Pisa (Fibonacci, about 1200 AD) is considered significant, which developed number sequences and the solution of equations by radicals with Diophantine equations.

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