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:
1100, 101200, 201300, ... 19998011999900, 19999012000000
There is only one such interval that does not contain any prime numbers, namely 16718011671900.
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.
A curiosity discovered while tackling problem 10
 Lord_Farin
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Re: A curiosity discovered while tackling problem 10
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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Re: A curiosity discovered while tackling problem 10
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.
Re: A curiosity discovered while tackling problem 10
Yes, though it is n!+2 up to n!+n, as n!+1 is not necessarily composite.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.
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