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across

  • 4 years ago

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

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  1. mathmate
    • 4 years ago
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    Does that have to do with log(n)/n and the Riemann-Zeta function?

  2. mathmate
    • 4 years ago
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    Oops, n/log(n)

  3. across
    • 4 years ago
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    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.

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

  5. across
    • 4 years ago
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    mathmate, your approximation is fairly close!

  6. across
    • 4 years ago
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    For corroboration, I wrote this: http://ideone.com/qoDbe I am now trying an analytical approach.

  7. mathmate
    • 4 years ago
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    Can you elaborate on your analytical approach, or is it proprietary?

  8. mathmate
    • 4 years ago
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    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.

  9. across
    • 4 years ago
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    That made me want to go back and recheck the code. ^^

  10. GT
    • 4 years ago
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    wow.....

  11. across
    • 4 years ago
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    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.

  12. GT
    • 4 years ago
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    Code looks good except comparison to 1.

  13. GT
    • 4 years ago
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    btw - why picture change?

  14. mathmate
    • 4 years ago
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    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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