

Prime Number Tendancy
Name: Francisco
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: N/A
Question:
As we can say there are 50% of odd numbers in a infinite
set of natural numbers, can we know the percentage of prime numbers which
exists on a infinite set of natural numbers?
For example:
The 590th prime number is 4297
So there is 13.731% of prime numbers between 1 and 4297
The 1437th prime number is 11981
So there is 11.994% of prime numbers between 1 and 11981
what happens with that percentage when we do the same as above with the
Nth prime number? Does it tend to zero? or does it tend to a constant?
Replies:
Francisco  no need to apologize for your English. You communicated your
question just fine. Unfortunately, I will not be able to communicate an answer
as easily.
The number of primes and their distribution, is not known. "Best guess" is
that the number of primes is infinite, but to my knowledge no such theorem
has been proven.
Not for having tried by a lot of mathematicians! There are various "sieves"
for primes. These are formulas for "catching" prime numbers. Mersenne primes
are an example of one such sieve. The famous mathematician, Gauss, presented
a formula for estimating the number of primes, p(n), less than an integer,
n: p(n)= n/ln(n) for large n. But there are many other such estimates, too.
There is a great deal of interest in primes because they are used in
cryptography  the study of secret messages. Many computer hours have been
spent examining whether there is any discernable pattern or frequency
distribution. I do not believe that a pattern has been extracted from the
vast number of known primes.
Vince Calder
Proofs as far back as Euclid (c. 300 BCE)
show that there are an infinite number of primes. For example: If
there were a finite number of primes you could claim there is a
largest. From that, you could take all of the primes and multiply
them together and add one to get a number N. N would not be
divisible by any of the primes that went into making it, and thus
itself would be prime and larger than the largest listed,
contradicting your statement that there was a finite number of primes.
Jake
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Update: June 2012

