Frequency of PrimesDate: 03/17/97 at 21:34:55 From: Jonah Knobler Subject: Frequency of Primes I am a student in an Algebra II class, and I'm wondering about number theory. I know what a prime number is, and I believe there are an infinite number of them (someone HAS proved this, right?) Is there any discernible pattern in the frequency of prime numbers? For instance, the distance between primes starts as: 1 (2->3) 2 (3->5) 2 (5->7) 4 (7->11) 2 (11->13) 4 (13->17) 2 (17->19) 4 (19->23) 6 (23->29) 2 (29->31). This sequence seems to have no pattern, but it is only a VERY small part of the sequence. My question is this: knowing that, as you continue toward greater primes, the distance between primes (usually) increases, is there a general trend in HOW this distance increases? Does it approximately follow a quadratic/logarithmic/linear/whatever function? Is it completely random? What does it look like on a graph? Seeing how important prime numbers are in number theory, I would assume that discovering an interesting number or formula popping up here (e.g. pi, e, etc.) would be fascinating. Thanks! - Jonah Knobler Date: 03/18/97 at 08:51:06 From: Doctor Rob Subject: Re: Frequency of Primes The Proof of the Infinite Number of Primes: This was done by Euclid more than 2000 years ago. He used a proof by contradiction. Suppose p(1), ..., p(n) is a list of all the prime numbers, i.e., that there are only finitely many. Then construct the number N = p(1)*p(2)*...*p(n) + 1. By the Fundamental Theorem of Arithmetic, this can be uniquely written as a product of prime numbers. If p(i) is one of those prime numbers, then N = p(i)*M, and so: 1 = N - p(1)*...*p(n) = p(i)*[M - p(1)*...*p(i-1)*p(i+1)*...*p(n)], which demonstrates that p(i) is a prime divisor of 1. This is, of course, impossible, since p(i) > 1. Thus none of the prime divisors of N appears in the list of primes. This contradicts the assumption that the list contained all primes. The conclusion is that no such list is complete, and the number of primes must be infinite. Is there a pattern for the distribution of prime numbers? This is a problem which has interested number theorists for several centuries. There is no discernible pattern known. Even such simple questions as, "Does 2 appear infinitely often?" are unresolved. This particular question is known as the "Twin Prime Conjecture." Some facts are known, however. For example, it is known that there is an infinite subsequence {x(n): n=1, 2, ...} of the prime numbers such that [x(2*n) - x(2*n-1)]/log_e[x(2*n)] is unbounded (gets bigger than any number you pick in advance). This (and more) was proved by Paul Erdos in 1949. >Seeing how important prime numbers are in number theory, I would >assume that discovering an interesting number or formula popping up >here (e.g. pi, e, etc.) would be fascinating. Yes, wouldn't it. Notice the appearance of e in the previous paragraph! Pi, e, and Euler's constant gamma frequently do appear in formulae or theorems involving prime numbers. For example, there is a theorem which says that if you pick two integers at random (in some sense which I won't go into), that the probability that they have no common prime divisor is 6/pi^2 ~=~ 0.6079271, which I think is really pretty! These are important and interesting questions in number theory, many of which are unsolved. It is heartening to see an Algebra II student with interest in them. Keep up the good questions! -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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