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    <p>Thanks for all of those answers. I suspected there was something fairly simple like that, but I couldn't find it at the time. I've done a bit more research too.</p> <p>As I want it for a <a href="http://en.wikipedia.org/wiki/Template:Number-theoretic_algorithms" rel="nofollow noreferrer">sieve</a> to generate the first <em>n</em> prime numbers, I want the approximation to be greater than or equal to the <em>n</em>th prime. (Therefore, I want the upper bound of the <em>n</em>th prime number.)</p> <p><a href="http://en.wikipedia.org/wiki/Prime_number_theorem#Approximations_for_the_nth_prime_number" rel="nofollow noreferrer">Wikipedia</a> gives the following upper bound for <code>n &gt;= 6</code></p> <pre><code>p_n &lt;= n log n + n log log n (1) </code></pre> <p>where <code>p_n</code> is the <em>n</em>th prime, and <code>log</code> is the natural logarithm. This is a good start, but it can overestimate by a not inconsiderable amount. <a href="https://www.maa.org/sites/default/files/jaroma03200545640.pdf" rel="nofollow noreferrer">This article</a> in <a href="http://www.maa.org/pubs/cmj.html" rel="nofollow noreferrer">The College Mathematics Journal</a> gives a tighter upper bound for <code>n &gt;= 7022</code></p> <pre><code>p_n &lt;= n log n + n (log log n - 0.9385) (2) </code></pre> <p>This is a much tighter bound as the following table shows</p> <pre><code>n p_n approx 1 error% approx 2 error% 1 2 10 29 31 6.90 100 541 613 13.31 1000 7919 8840 11.63 10000 104729 114306 9.14 104921 0.18 100000 1299709 1395639 7.38 1301789 0.16 1000000 15485863 16441302 6.17 15502802 0.11 10000000 179424673 188980382 5.33 179595382 0.10 </code></pre> <p>I implemented my <em>n</em>th prime approximation function to use the second approximation for <code>n &gt;= 7022</code>, the first approximation for <code>6 &lt;= n &lt; 7022</code>, and an array lookup for the first 5 prime numbers.</p> <p>(Although the first method isn't a very tight bound, especially for the range where I use it, I am not concerned as I want this for a sieve, and a sieve of smaller numbers is computationally cheap.)</p>
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    1. POIs there a way to find the approximate value of the nth prime?
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    1. USDennis
    1. USDavid Johnstone
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      1. VTUpMod
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    1. COUnfortunately, this formula appears to be broken. `p_8597` = 88789, but the formula gives 88759.3, which is an _underestimate_. It seems like it may be okay for n >= 8602.
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      1. USCharphacy
    2. COHow peculiar. That formula comes straight from [this paper](http://mathdl.maa.org/images/cms_upload/jaroma03200545640.pdf), which in turn is based on [this French paper](http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4242.pdf)...
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      1. USDavid Johnstone
    3. COFrom checking the primes up to 2e9 I got: `p_8621 = 89009, p ∈ [88746.2, 89014.4], delta ∈ [-0.9718, -0.9407]` By choosing 8621 you get the somewhat better approximation using the constant -0.9407. The next improvement is at `p_15957`. Yes, the paper must be wrong.
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