Historical Introduction to Number Theory : Part -3 #Number #theory

      Historical Introduction to Number Theory : Part -3     

                        ( part 2 continuation ) 

                          After Euclid in 300 BC no significant advances were made in  number theory until about AD 250 when another Greek mathematician,  Diophantus of Alexandria, published 13 books , six of which have been preserved. This was the first Greek work to make symmetric use of algebraic symbols. Although his algebraic notation seems awkward by present – day standards, Diophantus was able to solve certain algebraic equations involved two or three unknowns. Many of his problems originated from number theory and it was natural for him to seek integer solutions of equations. Equations to be solved with integer values of the unknowns are now called Diophantine equations, and the study of such equations is known as Diophantine analysis. The equation x²+y²=z² for Pythagorean triples is an example of a Diophantine equation.                           

                       After Diophantus, not much progress was made in the theory of numbers until the seventeenth century, although there is  some evidence that the subject began to flourish in the Far East ---- especiallly in India- in the period between AD 500 an AD 1200.

                        In the 17^th century the subject was revived in western Europe , largely through the efforts of a remarkable French mathematician , Pierre de Fermat (1601 -  1665), who is generally acknowledged to be the father of modern number theory. Fermat derived much of his inspiration from the works of Diophantus . he was the first to discover really deep properties of the integers. For example, Fermat proved the following surprising theorems-

                        “ Every integer is either a triangular number or a sum of two or three triangular numbers “

                        “ every integer is either a square or a sum of two, three or four squares. “

                        “ Every integer is either a pentagonal or the sum of two, three, four or five pentagonal numbers” and so on.

                       Fermat also discovered that every  prime number of the form 4n+1 such as 5,13,17,29,37,41 etc., is a sum of two squares. For example

        5= 1²+2²,

       13=2²+3²,

      17=1²+4²,

      29=2²+5²,

      37=1²+6²,

      41=4²+5².

                        Shortly after Fermat’s time, the names of Euler (1707-1783), Lagrange(1736-1813), Legendre (1752-1833), Gauss ( 1777-1855), and Dirichlet (1805-1859)  became prominent in the further development of the subject. The first textbook in number theory was published by Legendre in 1798. Three years later Gauss published Disquisitions Arithmeticae,  a book  which transformed the subject into systematic and beautiful science. Although he made a wealth of contributions to other branches of mathematics, as well as to other sciences,  Gauss himself considered his book on number theory to be his greatest work.

                        In the last 100 years or so since Gauss’s time there has been an intensive development of the subject in many  different directions. It would be impossible to give in a few pages a fair cross-section of the types of problems that are studied in the theory of  numbers. The field is vast and some parts require a profound knowledge of higher mathematics. Nevertheless, there are many problems in number theory which are very easy to state. Some of these deal with prime numbers, and we devote the rest of this introduction to such problems.

                       The primes less than 100 have been listed above. A table listing all primes less than 10  million  was published in 1914 by an American mathematician , D. N. Lehmer . There are exactly 664,579 primes less than 10 million , or about 6 ½ % . More recently the son of D.N. Lehmer , D.H. Lehmer calculated the total number of primes less than 10 billion; there are exactly 455,052,512 such primes  or about 4 ½ % although all these primes are not known individually    .

                        A close examination of a table of primes reveals that they are distributed in a very irregular fashion. The tables show long gaps between primes. For example, the prime 370,261  is followed by 111 composite numbers. There are no primes between 20,831,323 and 20,831,533 .  It is easy to prove that arbitrary large gaps between prime numbers  must eventually occur.

***Historical Introduction to Number Theory : Part -4***

 #Number #theory