Historical Introduction to Number Theory : Part -5 #number #theory
Historical Introduction to Number Theory : Part -5
( part 4 continuation )
In 1851 the Russian mathematician Chebyshev made an important
step forward by proving that if the ratio 𝛑(x ) log x / x
did tend to a limit , then this limit must be 1. However he was unable
to prove that the ratio does tend to a limit.
In 1859 Riemann
attacked the problem with analytic methods, using a formula discovered by Euler
in 1737 which relates the prime numbers
to the function
ζ (s)= Σ 1/ns for a real s >1.
Riemann consider complex values of small s and outlined an
ingenious method for connecting the distribution of primes to properties of the
function ζ (s). The mathematics needed to justify all the details of
his method had not been fully developed and Riemann was unable to completely settle the problem before his death
in 1866.
30 years later the necessary
analytic tools were at hand and in 1896 J. Hadamard and C.J.de la vallee Poussin independently and almost simultaneously succeeded in
proving that
Lim 𝛑(x
) log x / x =1 as x tends to ∞.
This remarkable result is called the PRIME
NUMBER THEOREM, and its proof was one of
the crowning achievement of analytic number theory.
In 1949, two contemporary mathematicians,
Atle Selberg and Paul Erdos caused a sensation in the mathematical world when they discovered an elementary proof of
the prime number theorem. Their proof,
though very intricate, makes no use of ζ (s) nor
of complex function theory and in principle is accessible to anyone familiar
with elementary calculus.
One of the most famous problems
concerning prime numbers is the so – called Goldback conjecture wrote to Euler suggesting
that every even number ≥4 is a sum of two primes. For example
4= 2+2,
6= 3+3, 8=3+5, 10= 3+7 = 5+5, 12= 5+7 etc.
This conjecture is undecided to
this day, although in recent years some progress has been made to indicate that
it is probably true. Now why do mathematicians think it is probably true if
they haven’t been able to prove it? First of all , the conjecture has been verified by actual
computation for all even numbers less than 33 x 106 . It
has been found that every even number
greater than 6 and less than 33
x 106 is, in fact, not only
the sum of two odd primes but
the sum of two distinct off primes. But in
number theory verification of a few thousand cases is not enough evidence to
convince mathematicians that something
is probably true. For example, all the odd primes fall into two
categories, those of the form 4n+1 and those of the form 4n+3 . let 𝛑1
(x) denote all primes ≤ x that are of the form 4n+1, and let 𝛑3 (x ) denote the number that are of
the form 4n+3 . It is known that there
are infinitely many primes of both
types. By computation it was found that 𝛑1 (x) ≤ 𝛑3 (x
) and also infinitely many x for which 𝛑3 (x ) ≤ 𝛑1
(x). conjectures about prime numbers can be erroneous even if they are verified
by computation in thousands of cases.
Therefore, the fact that Goldbach’s
conjecture has been verified for all even numbers less than 33 x 106
is only a tiny bit of evidence in its favor.
Another way that mathematicians
collect evidence about the truth of a particular conjecture is by proving other
theorems which are somewhat similar to the conjecture. For example, in 1930 the
Russian mathematician Schnirelmann proved that there is a number M such that
every number n from some point on is a sum
of M or fewer primes :
n= p1+p2+…+pM
( for sufficiently large n).
if we knew that M were equal to 2
for al even n, this would prove Goldbach’s conjecture for all sufficiently
large n. in 1956 the Chinese mathematician Yin Wen- Lin proved that M ≤ 18.
That is , every number n from some point on is a sum of 18 or fewer primes. Schnirelmann’s result is considered
a giant step toward a proof of Goldback’s conjecture. It was the first real
progress made on this problem in nearly 200 years.
A much closer approach to a
solution of Goldbach’s problem was made in 1937 by another Russian
mathematician, I. M. Vinogradov, who proved that from some point on every odd
number is the sum of three primes :
n = p1+p2+p3 ( for sufficiently large n , n is odd ).
In fact, this is true for all odd
n greater than 33^15. To date
, this is the strongest piece of evidence in favor of Goldbach’s conjecture. For
one thing , it is prove that Vinogradov’s
theorem is a consequence of Goldbach’s statement. That is, Goldbach’s
conjecture is true , then it is easy to deduce Vinogradov’s statement. The big
achievement of Vinogradov was that he was able to prove his
result without using Goldbach’s statement. Unfortunately, no one has
beenable to work it the other way around and prove Goldbach’s statement from
Vnogradov’s.
Another piece of evidence in favor of Goldbach’s conjecture was found in
1948 by the Hungarian mathematician Renyi who proved that there Is a number M
such that every sufficiently large even number n can be written as a prime plus
another number which has no more than M
prime factors :
Nice
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