Loading ...
Sorry, an error occurred while loading the content.

Are Big Numbers More Likely To Be Prime?

Expand Messages
  • Chester Elders
    I have no idea whether this proof is unique to me; but, I think I can prove that the probability that a natural number is prime is directly proportional to
    Message 1 of 4 , May 23, 2010
    • 0 Attachment
      I have no idea whether this proof is unique to me; but, I think I can prove that the probability that a natural number is prime is directly proportional to it's size.

      Proof:

                Let A be a natural number. Let S be the set of all prime numbers such that any element of S is less than the step function of the square root of A. Define the probability that A is a prime number as the quotient (A-S)/A. Since the quotient of the square root of a natural number divided by the number decreases in proportion to the size of that number, it follows that the limit of (A-S)/A approaches 1 as A gets arbitrarily large. This is tantamount to saying that the probability that A is prime increases as A increases. QED.

               Wouldn't this imply that the prime numbers get arbitrarily dense when one considers extremely large natural numbers? What really has me worried is the seemingly inescapable conclusion that, for sufficiently large natural numbers, wouldn't primes tend to become sequential? I know that's not possible because the distribution of prime numbers is random. I assume that probability holds uniformly across the number line.

               I would appreciate some feedback.

               Sincerely,
               Chester 




      [Non-text portions of this message have been removed]
    • Alan Eliasen
      ... Chester, You appear to not know about one of the best-known facts about the distribution of prime numbers: the probability that a randomly-chosen positive
      Message 2 of 4 , May 23, 2010
      • 0 Attachment
        On 05/23/2010 03:17 AM, Chester Elders wrote:
        > I have no idea whether this proof is unique to me; but, I think I can
        > prove that the probability that a natural number is prime is directly
        > proportional to it's size.

        Chester,

        You appear to not know about one of the best-known facts about the
        distribution of prime numbers: the probability that a randomly-chosen
        positive integer N is prime is approximately equal to 1/ln[N] where ln
        is, of course, the natural log function.

        --
        Alan Eliasen
        eliasen@...
        http://futureboy.us/
      • Chris Caldwell
        ... increases as A increases. QED. Alan pointed out the Prime Number Theorem, which is truly important, but also consider common sense: past 2, half of the
        Message 3 of 4 , May 23, 2010
        • 0 Attachment
          > This is tantamount to saying that the probability that A is prime
          increases as A increases. QED.

          Alan pointed out the Prime Number Theorem, which is truly important, but
          also consider common sense: past 2, half of the numbers composites
          divisible by 2. Past 3, half of those remaining numbers are divisible
          3, past 5, 1/5 are divisible by 5... obviously the density is going down
          (on the average).

          Good luck!
        • Peter Kosinar
          ... So far okay... ... ... but why exactly would this ratio describe the probability of a number being a prime? Peter
          Message 4 of 4 , May 23, 2010
          • 0 Attachment
            > Let A be a natural number. Let S be the set of all prime numbers such
            > that any element of S is less than the step function of the square root
            > of A.

            So far okay...

            > Define the probability that A is a prime number as the quotient (A-S)/A.

            ... but why exactly would this ratio describe the probability of a number
            being a prime?

            Peter
          Your message has been successfully submitted and would be delivered to recipients shortly.