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 infinitas". 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 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:

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

See also

• Goldbach's conjecture

References

• Pelegrí Viader, Lluís Bibiloni, Jaume Paradís. "On a series of Goldbach and Euler" (PDF). {{cite journal}}: Cite journal requires |journal= (help)CS1 maint: multiple names: authors list (link)
• Graham, Ronald (1988). Concrete Mathematics. Addison-Wesley. ISBN 0201142368. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)