6431
/www/nrich/html/content/id/6431/
1
2011-06-09T11:35:54
<mdoxml xmlns:ns0="http://nrich.maths.org/mdo" version="1.0">
<br />
<ul id="buttonBar">
<li>
<a href="http://nrich.maths.org/396&part=">Warm-up
problem</a>
</li>
<li>
<a href="http://nrich.maths.org/7029&part=">Try this next</a>
</li>
<li>
<a href="https://nrich.maths.org/discus/messages/27/27.html?1307565044">
Ask NRICH</a>
</li>
<li>
<a href="http://en.wikipedia.org/wiki/Prime-counting_function">Read
all about it</a>
</li>
<li>
<a href="http://nrich.maths.org/6322&part=solution">Last week's
solution</a>
</li>
</ul>
<div>
<br />
<br />
The <span style="font-style: italic;">prime counting
function</span> $\Pi(x)$ counts how many prime numbers are less
than or equal to $x$ for any positive value of $x$. Since the
primes start $2, 3, 5, 7, 11, 13, \dots$ we therefore have, for
example, $\Pi(11) = 5$ and $\Pi(8) = 4$.<br />
<br />
It is believed by mathematicians that $\frac{x}{\ln(x)}$ is a good
approximation to $\Pi(x)$. It is believed to get progressively
better as $x$ increases to very, very large numbers. How well does
it work for lower values of $x$ (up to the 100,000th prime)<br />
<br />
Use the following interactivity to examine the percentage accuracy
of this approximation for these values.<br />
<br />
<br />
<mdo:flash height="375" width="500">
<param name="movie" value="/content/id/6431/PrimeLookup.swf" />
<param name="flashplayerversion" value="9" />
</mdo:flash>
<br />
<br />
Use a few sensible values / choice of axes to try to create a
useful graphical representation of $\ln(\Pi(x))$ against $\ln(x)$
for $x$ taking values up to about a million. Use your curve to try
to predict $\Pi(x)$ for a few values of $x$ away from your data
points. How close are your estimates for whole number multiples of
$100,000$?<br />
<br />
Use your judgement to try to extrapolate your curve to make
approximations as to<br />
$$<br />
\Pi(10^7)\quad\quad \Pi(10^8)\quad\quad \Pi(10^9)<br />
$$<br />
<br />
<br />
<br />
</div>
<div class="framework">
<span style="font-style: italic;">Did you
know ... ?</span>
<br />
<br />
Although prime numbers are distributed with some large-scale
regularity across the natural numbers, as this problem indicates,
there is no known process which generates the prime
numbers. </div>
<br />
</mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Here are some low values<br></br>
<br></br>
<mdo:image height="179" width="287" src="tables%20of%20values.JPG" alt=""></mdo:image><br></br>
<br></br>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Here are some low values<br></br>
<br></br>
<mdo:image height="179" width="287" src="tables%20of%20values.JPG" alt=""></mdo:image><br></br>
<br></br>
<br></br>
<br></br></mdoxml>
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Weekly challenge 44: Prime counter
A weekly challenge concerning prime numbers.
Collections
Weekly Challenge
Numbers and the Number System
Prime numbers
Information and Communications Technology
Interactivities
Sequences, Functions and Graphs
Logarithmic functions