## A curiosity discovered while tackling problem 10

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perfidius
Posts: 1
Joined: Mon Jul 23, 2012 1:06 am

### A curiosity discovered while tackling problem 10

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.
Lord_Farin
Posts: 239
Joined: Wed Jul 01, 2009 10:43 am
Location: Netherlands

### 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.
pimspelier
Posts: 41
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Location: The Netherlands

### 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.
jaap
Posts: 555
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### Re: A curiosity discovered while tackling problem 10

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.