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### Re^4: a close prime number

by Roy Johnson (Monsignor)
 on Feb 11, 2005 at 21:12 UTC ( #430279=note: print w/replies, xml ) Need Help??

in reply to Re^3: a close prime number
in thread a close prime number

You can do better than using the arbitrary range of +100/-100, (which isn't going to be good enough when your numbers get larger).
Really? Is there a range of 200 consecutive numbers within which there are no primes? My understanding was that they appeared with regular density, no matter how high you climbed.

Update:
And I get to answer my own question. I Googled up this webpage. It's 210 numbers to the next prime from 20831323.

Caution: Contents may have been coded under pressure.

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Re^5: a close prime number
by Anonymous Monk on Feb 12, 2005 at 20:51 UTC
My understanding was that they appeared with regular density
Regular density? Depending on what you mean by "regular" .. There is the famous prime number theorem. If π(n) is the number of primes less than n, then π(n) tends to n/ln(n) as n grows to infinity. Thus you could say that the density if primes within {1,..,n} is ln(n)/n. But this density gets smaller and smaller as n gets larger, thus the primes have to be getting (on average) farther and farther apart.

Also, as blazar mentions below. If you want to find a number X such that the first prime after X is at least X+200, just let X be 201! (that's with a factorial). Now X+1 may be prime, but X+2 is not prime since 2 divides X and 2. X+3 is not prime since 3 divides X and 3, etc... all the way until X+201.

Re^5: a close prime number
by blazar (Canon) on Feb 12, 2005 at 10:53 UTC
Really? Is there a range of 200 consecutive numbers within which there are no primes? My understanding was that they appeared with regular density, no matter how high you climbed.
Huh?!? You're joking too, arent' you? Can you find many prime numbers amongst the numbers q+2, q+3, ..., q+p, where q is the product of all primes up to p? This is so elementary that it is IIRC on the second or third page of "Hardy and Wright"...

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