Goldbach–Euler Theorem

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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 "*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:<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

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

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

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

This result was first published in Euler's 1737 paper "

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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