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September 18th, 2013, 03:40 PM   #1
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Number of primes in an interval

For how many nonnegative integers n?25 does there exist a set of 1000 consecutive positive integers containing exactly n primes?
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September 18th, 2013, 04:11 PM   #2
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Re: Number of primes in an interval

I imagine this is true for all 26. It's true for 0 by the prime number theorem; let me hunt around for some numerics.

Edit: It's also true for 1 by the prime number theorem. Otherwise, all primes would have a gap of less than a thousand, which would give them density >= n/998 which is much bigger than n/log n asymptotically.
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September 18th, 2013, 04:44 PM   #3
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Re: Number of primes in an interval

The first example for n = 0 is {1693182318746372, ..., 1693182318747371}.

I don't know when the first occurrence is for n = 1, but one example is {1693182318746352, ..., 1693182318747351}. The first example has all terms greater than 3*10^11.

I don't know when the first occurrence is for n = 2, but one example is {1693182318746310, ..., 1693182318747309}. The first example has all terms greater than 3*10^9.

I don't know when the first occurrence is for n = 3, but one example is {1686994940955704, ..., 1686994940956703}. The first example has all terms greater than 4*10^8.

...

The first example for n = 25 is {172154114, ..., 172155113}.
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September 18th, 2013, 04:59 PM   #4
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Re: Number of primes in an interval

This is a nice question , i like it.

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