Historical Introduction to Number Theory : Part -4 #number #theory

          Historical Introduction to Number Theory : Part -4                                                                  ( part 3 Continuation )

     On the other hand , the tables indicate that consecutive primes, such as 3 and 5 or 101 and 103 keep recurring. Such pairs of primes which differ only by 2 are known as twin primes. There are over 1000 such pairs below 100,000 and 8000 below 1,000,000. The largest pair known to date is 76 . 3¹³⁹ -1 and   76 . 3¹³⁹ +1. Many mathematicians think there are infinitely many such pairs, but no one has been able to prove this as yet.

 One of the reasons for this irregularity in distribution of primes is that no simple formula exists for producing all the primes.  Some formulae do yield many primes. For example, the expression x²-x+41 gives a prime for x= 0,1,2,…40, where as x² -79x +1601 gives a prime for x= 0,1,2…79. However , no such simple formula can give a prime for all x , even if cubes and higher powers are used. In fact, in 1752 Goldbach proved that no polynomial in x with integer coefficients can be prime for all x, or even for all sufficiently large x.

Some polynomials represent infinitely many primes. For example, as x runs through the integers 0,1,2,3,… , the linear polynomial 2x+1 gives all the odd numbers hence infinitely many primes. Also , each of the polynomials 4x+1 and 4x+3 presents infinitely many primes. In a famous memoir published in 1837, Dirichlet proved that , if a and b are positive integers with no prime factors in common, the polynomial ax+b gives infinitely many primes as x runs through all the positive integers. This result is now known as Dirichlet’s theorem on the existence of primes in a given arithmetical progression.

To prove this theorem, Dirichlet went outside the realms of integers and introduced tools of analysis such as limits and continuity. By so doing he laid the foundations for a new branch of mathematics called analytic number theory, in which ideas and methods of real and complex analysis are brought to bear on problems about that integers.

It is not known if there is any quadratic  polynomial ax²+bx+c  with a≠ 0 , which represents infinitely many primes. However , Dirichlet used his powerful analytic methods to prove that , if a, 2b, c have no prime factor in common, the quadratic polynomial in two variables ax²+abxy+cy² represents infinitely many primes such as x and y  runs through the positive integers.

Fermat thought that the formula 2^{2^n} +1 would always give prime for n=0,1,2,… . these numbers are called Fermat numbers and are denoted by F_n . The first five are

F₀ = 3, F₁ = 5, F₂ = 17, F₃ = 257, F₄= 65,537, and they are all primes. However , in 1732 Euler found that F₅ was composite; in fact, F₅ = 2³² +1 = (641)(6,700,417).

These numbers are also of interest in plane geometry. Gauss proved that if F_n is a prime, say F_n =p , then a regular polygon of p sides can be constructed with straightedge and compass.

Beyond F₅ , no further Fermat primes have been found. In fact, 5 ≤ n ≤ 16 each Fermat number F_n is composite. Also , F_n is known  to be composite for the following further isolated values of n;

n= 18,19,21,23,25,26,27,30,32,36,38,39,42,52,55,58,63,73,77,81,117,125,144,150,207,226,228,260,267,268,284,316,452 and 1945.

The greatest known Fermat composite F₁₉₄₅ , has more than 10⁵⁸² digits, a number larger than the number of letters in the Los Angles and New York telephone directories combined .

It was mentioned earlier that there is no simple formula that gives all the primes. In this connection, we should mention a result discovered in 1947 by an American mathematician , W.H. Mills. He proved that there is some number A, greater than 1 but not an integer, such that [A^{3^x}] is prime for all x=1,2,3,….

Here [A^{3^x} ] means the greatest integer ≤ A^{3^x} . unfortunately no one knows what A is equal to .

The foregoing results illustrates the irregularity of the distribution of the prime numbers. However, by examining large blocks of primes one finds that their average distribution seems to be quite regular. Although there is no end to the primes, they become more widely spaced , on the average , as we go further and further in table. The question of the diminishing frequency of primes was subject of much speculation in the early 19 ^th century. To study this distribution, a function , denoted by 𝛑(x), which counts the number of primes ≤x. Thus 

𝛑(x) = the number of primes p satisfying 2 ≤ p ≤ x. 

***Historical Introduction to Number Theory : Part -5***

  #number #theory