I posted this to Dr. Math but I'm too excited to wait for their response.(adsbygoogle = window.adsbygoogle || []).push({});

OK, so start with the following equation, http://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_…#Summability" by Ramanujan and Euler:

1 + 2 + 3 + 4 + ... = -1/12

Weird, yes, but there are http://planetmath.org/encyclopedia/PAdicValuation.html" [Broken] under which these kinds of expressions are meaningful.

OK, so let's call the series s:

s = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + ...

Now, we can gather all multiples of 2 into one term:

s = 1 + 2s + 3 + 5 + 7 + 9 + 11 + ...

Similarly, we can gather all multiples of 3:

s = 1 + 2s + 3s + 5 + 7 + 11 + ...

Continue sieving this way in a manner similar to Eratosthenes until you have:

s = 1 + s * ( sum{all primes p} p )

Rearranging:

(s - 1) / s = sum{all primes p} p

But since we already know the value of s = -1/12:

(-13/12) / (-1/12) = sum{all primes p} p

sum{all primes p} p = 13

QED

I see no flaw in my reasoning.

Clayton

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# Is the sum of all primes = 13?

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