Goldbach–Euler theorem

In mathematics, the Goldbach–Euler theorem (also known 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:

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

Goldbach's original proof to Euler involved assigning a constant to the harmonic series: ${\displaystyle \textstyle x=\sum _{n=1}^{\infty }{\frac {1}{n}}\ }$, which is divergent. Such a proof is not considered rigorous by modern standards. It is also interesting to note that there is a strong resemblance between the method of sieving out powers employed in his proof and the method of factorization used to derive Euler's product formula for the Riemann zeta function.

Let x be given by

Since the sum of the reciprocal of every power of two is ${\displaystyle \textstyle 1={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots }$, subtracting the terms with powers of two from x gives

...
Wikipedia