Goldbach-Euler theorem

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In mathematics, the Goldbach-Euler theorem (also know as Goldbach's theorem), states that the sum of 1/(p − 1) over the set of perfect powers p, excluding 1 and omitting repetitions, converges to 1:

\sum_{p}\frac{1}{p-1}= {\frac{1}{3} + \frac{1}{7} + \frac{1}{8}+ \frac{1}{15} + \frac{1}{24} + \frac{1}{26}+ \frac{1}{31}}+ \cdots = 1.

This result was first published in Euler's 1737 paper "Variae observationes circa series infinitas". Euler attributed the result to a letter (now lost) from Goldbach.

[edit] Proof

Goldbach's original proof to Euler involved assigning a constant to the harmonic series: x = \sum_{n=1}^\infty \frac{1}{n}\ , which is clearly divergent.

Such a proof may be considered as "not rigorous" by modern standards.

It can be shown that the sum of 1/p over the set of perfect powers p, excluding 1 but including repetitions, converges to 1 as well:

\sum_{p}\frac{1}{p} = \sum_{m=2}^\infty \sum_{n=2}^\infty \frac{1}{m^n} = 1.

[edit] See also

[edit] References