ABSTRACT
Number theory is one of the oldest branches of mathematics and deals principally with properties of the integers, the simplest kinds of numbers. The modern system used to write numbers down is quite different and is called positional number system. It seems to have been in place by ninth century in India. Its genius is that only 10 symbols, called digits. In 1971, Yuri Matiyasevich found a polynomial in 26 variables of degree 25 with property that when non-negative integers are substituted for the variables the positive values it takes are all and only the primes. Prime numbers play an important role in exchanging secret information. The Fibonacci numbers also have an unexpected connection with Euclid's algorithm. It is a theorem of Gabriel Lame that the number of divisions needed in the application of Euclid's algorithm is less than or equal to five times the number of digits in the smaller of the two numbers whose gcd is being calculated.
