A curiosity discovered while tackling problem 10

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perfidius
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A curiosity discovered while tackling problem 10

Post by perfidius »

While tackling problem 10 (summing all the prime numbers that are less than 2000000) in what was probably a very strange manner, my workings uncovered this potentially interesting fact:

If you divide 2000000 into intervals of 100 (or is it 99? These things confuse me) like so:
1-100, 101-200, 201-300, ... 1999801-1999900, 1999901-2000000
There is only one such interval that does not contain any prime numbers, namely 1671801-1671900.

Shocking stuff eh?
I'm not sure if I was expecting there to be more or no intervals without any primes, but I was definitely surprised.
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Lord_Farin
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Re: A curiosity discovered while tackling problem 10

Post by Lord_Farin »

It has already been shown that arbitrarily large prime gaps (difference between consecutive primes) exist. This is stronger observation implies that there are very many intervals of length 100 without a prime.
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pimspelier
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Re: A curiosity discovered while tackling problem 10

Post by pimspelier »

It is interesting: you can easily make a prime gap of any length: if you take n!, n!+2,n!+2...,n!+n then all of those are not prime.
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jaap
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Re: A curiosity discovered while tackling problem 10

Post by jaap »

pimspelier wrote:It is interesting: you can easily make a prime gap of any length: if you take n!, n!+2,n!+2...,n!+n then all of those are not prime.
Yes, though it is n!+2 up to n!+n, as n!+1 is not necessarily composite.
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