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Name:  Francisco
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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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