Goldbach–Euler theorem

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 infinita"s". Euler attributed the result to a letter (now lost) from Goldbach.

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 divergent. Such a proof is not considered 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:

:$sum\_\{p\}frac\{1\}\{p\}\; =\; sum\_\{m=2\}^infty\; sum\_\{n=2\}^infty\; frac\{1\}\{m^n\}\; =\; 1.$

ee also

* Goldbach's conjecture

References

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