Historical Introduction to Number Theory : Part -2 #number #theory #historical #background

 Historical Introduction to Number Theory : Part -2

                ( part 1 continuation )

                                                        

The following are few examples of Pythagorean triples

(3,4,5), (5,12,13), (7,24,25), (9,40,41), (11,60,61), (13, 84,85), (15,112,113),(17,144,145) (19,180,181) etc.

There are other Pythagorean triples besides these ; for example : (8,15,17), (12,35,3),(16,63,65) (20,99,101) etc.

In these examples we have z=y+2.

Plato found a method for determining all these triples ; in modern notation they are given by the formulae

X=4n,  y= 4n²-1,  z= 4n² +1.

Around 300 BC an important event occurred in the history of mathematics. The appearance of Euclid’s Elements, a collection of 13 books , transformed mathematics from numerology into a deductive science. Euclid was the first to present mathematical facts along with rigorous proofs of these facts.

Three of the 13 books were devoted to the theory of numbers . In book IX Euclid proved that there are infinitely many primes. His proof is still taught in the classroom today. In book X he gave a method for obtaining all Pythagorean triples although he gave no proof that his method did, indeed, give them all. The method can be summarized by the formulae x= t( a²-b²) ,  y- 2tab,   z= t(a²+b²), where t ,a, are arbitrary positive integers such that  a >b , a and b have no prime factors in common, and one of a or b is odd, the other even. Euclid also made an important contribution to another problem posed by the Pythagoreans – that of finding all perfect numbers.

  The number 6 was called a perfect number, because 6= 1+2+3, the sum of all its proper divisors ( i.e. the sum of all divisors less than 6 ) . Another example of a perfect number is 28 because 28= 1+2+4+7+14, and 1, 2,4,7 ,and 14 are the divisors of 28 less than 28.  The Greeks referred to the proper divisors of a number as its “  parts “ . They called 6 and 28 perfect numbers because in each case the number is equal to the sum of all its parts . in book IX, Euclid found all even perfect numbers. He proved that an even number is perfect if it has the form 2^p-1 ( 2^p-1) ,

                      Where both p and 2^p -1 are primes.

2000  years later , Euler proved the converse of Euclid’s theorems. That is , every even perfect number must be of Euclid’s type. For example , for 6 and 28  we have

               6= 2^2-1(2²-1) = 2.3 and 28= 2^3-1( 2³-1)= 4.7.

The first five even perfect numbers are

 6, 28, 496, 8128 and 33,550,336.

Perfect  numbers are very rare indeed. At the present time (1975) only 24 perfect numbers are known. They correspond to the following values of p in  Euclid’s formula:

       2,3,5,7,13,17,19,31,61,89,107, 127,521,607,1279,2203,2281,3217,4253,4423,9689,9941,11213, 19937.

 Numbers of the form 2^p -1, where p is prime, are now called Mersenne numbers and are denoted by M_p, in honor of Mersenne , who studied them in 1644. It is known that M_p is prime for the 24 primes listed above and composite for all other values of p < 257, except possibly for

        P= 157, 167,193,199,227,229;

For these it is not yet known whether M_p is prime or composite. No odd perfect numbers are known; it is not even known if any exist. But if any do exist they must be very large; in fact , greater than  10⁵⁰.

We turn now to a brief description of the history of the theory of numbers since Euclid’s time. 

***Historical Introduction  to Number Theory :  Part 3***