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I am asking myself if instead of working with the primes in the calculation of $\pi(x)$ up to $x$, we instead work with the composite numbers and then using a simple subtraction to get $\pi(x)$. After all it must be much easier to deal with the composite numbers. We only need to look at $x/3$ to get $\pi(x)$ since $2/3$ of the numbers are multiples of $2$ and $3$. So we can write $c(x)+\pi(x) = x/3$ ( and add the $2$ coming from primes $2$ and $3$ ) to get the correct result ( $c(x)$ being of course the number of composite up to $x/3$ not included the multiples of $2$ and $3$ of course).

We know how to produce the composite numbers, but we don't know if a given number is a prime without testing it. What would be wrong with that?

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    @simplicity Watch your etiquette: http://math.stackexchange.com/faq#etiquette . If you think the question doesn't merit attention, then downvote it. But don't insult people.2011-11-13
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    @simplicity That's just rude. I've seen you throwing around words like "stupid", "dumb" and "silly" in this and other commentary. Do everyone a favor and save that denigrating sort of language for somewhere else.2011-11-13
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    Generally to see if something is prime or not isn't even that hard. It can be done in polynomial time http://en.wikipedia.org/wiki/AKS_primality_test2011-11-13
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    @Mbel, this site does not work well if you use it to suggest new approaches to problems. If you think you have a useful and promising approach to something, then carry it out. «What would be wrong with that?» is not a real question as far as math.stackexchange goes.2011-11-23

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