How many prime numbers are there between 1,000,000 and 1,100,000?

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- across

How many prime numbers are there between 1,000,000 and 1,100,000?

- schrodinger

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- mathmate

Does that have to do with log(n)/n and the Riemann-Zeta function?

- mathmate

Oops, n/log(n)

- across

With this question, I am seeking to clash a mathematical approach with a computerized one. Although it would be interesting to see in how many different ways we can tackle this exercise.

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## More answers

- mathmate

So that makes 6692.7, say 6693.
Is there a better estimate using R-Z function, I don't really understand it.

- across

mathmate, your approximation is fairly close!

- across

For corroboration, I wrote this: http://ideone.com/qoDbe
I am now trying an analytical approach.

- mathmate

Can you elaborate on your analytical approach, or is it proprietary?

- mathmate

6693 wasn't even close, you were being polite!
Using the "offset logarithmic integral"
\[Li(n) = \int\limits_{2}^{n}\ \frac{dt}{\log(t)}\]
I get 7212.99 > 7213.
Using your code, I get 7216.
So now it's getting close.
Also, your code included 1 as a prime, which it is not. This does not change the counts over 1 though.

- across

That made me want to go back and recheck the code. ^^

- anonymous

wow.....

- across

Interesting. Using the logarithmic integral from 2,000,000 to 2,100,000 you get 6881, whereas the (fixed) code chunks out 6871.
The error is now negative.

- anonymous

Code looks good except comparison to 1.

- anonymous

btw - why picture change?

- mathmate

I get the same answers, except that I get 6872 for the actual count, both using your code and my code.

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