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:

<math>sum_frac= + cdots = 1.</math>

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:<math> textstyle x = sum_^infty frac </math>, 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

<math>x = 1 + frac + frac + frac + frac + frac + frac + frac cdots</math>

Since the sum of the reciprocal of every power of two is <math> textstyle 1 = frac + frac + frac + frac + cdots</math>, subtracting the terms with powers of two from x gives

<math>x - 1 = 1 + frac + frac + frac + frac + frac + frac + frac + cdots</math>

Repeat the process with the terms with the powers of three: <math>textstyle frac = frac + frac + frac + frac + cdots</math>

<math>x - 1 -......

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