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Friday, April 9, 2010

Prime numbers

I find the following identity rather intriguing...
$\sum_{k=1}^\infty\frac{1}{k} = \prod_{p \in P}\frac{1}{1-\frac{1}{p}}$.

The LHS of the equation is a sum over all positive integers $1,2,3 \dots$, the RHS of the equation is a product over all the prime numbers.

The prime numbers is the set of all integers which are (only) divisible by 1 and itself. There is no closed formula which generates the primes 2,3,5,7,11,13,17,19,23,29, ... There are approximately x / log x primes less than x.

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