Since the sum of the reciprocal of every power of two is , subtracting the terms with powers of two from x gives
Repeat the process with the terms with the powers of three:
Absent from the above sum are now all terms with powers of two and three. 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 one. By subtracting the previous equation from the definition of x given above, we obtain
where the denominators now consist only of perfect powers minus one.
While lacking mathematical rigor, Goldbach's proof provides a reasonably intuitive visualization of the problem. 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 over the set of perfect powers p, excluding 1 but including repetitions, converges to 1 by demonstrating the equivalence: