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:
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.
Continue by removing terms with powers of 5, 6 and so on until the right side is exhausted to the value of 1.
Eventually, we obtain the equation which we rearrange into where the denominators consist of all positive integers that are the non-powers minus 1.
given above, we obtain where the denominators now consist only of perfect powers minus 1.
While lacking mathematical rigor, Goldbach's proof provides a reasonably intuitive argument for the theorem's truth.
Rigorous proofs require proper and more careful treatment of the divergent terms of the harmonic series.
Other proofs make use of the fact that the sum of 1/(p − 1) over the set of perfect powers p, excluding 1 but including repetitions, converges to 1 by demonstrating the equivalence: