The Goldbach-Euler Theorem: Why the Sum of Reciprocals of Perfect Powers Equals 1

Perfect Powers and a Curious Question

What makes mathematics so fascinating? Surprising facts turn up everywhere you look.

This past weekend we were working on a problem about integers less than 1000 that are perfect powers of other integers — numbers like 4, 8, 9, 16, 25, 27, and so on. These are called perfect powers.

Now here’s a question: what happens if you sum the series of reciprocals of each perfect power minus one — excluding 1 itself and any duplicates?

Computing the Sum of Reciprocals of Perfect Powers

If we add up 40 such terms up to 1000, the result is roughly 0.96. But what does the sum converge to as we go to infinity? This is where things get beautiful.

It turns out the sum equals exactly 1. This result is known as the Goldbach-Euler theorem. In the 1730s, Christian Goldbach outlined the solution in a letter to Leonhard Euler, and in 1737 Euler published a paper with a full treatment.

The Goldbach-Euler Theorem and Its Flawed Original Proof

There’s a catch, though. Goldbach and Euler’s solution, while arriving at the correct answer, was actually flawed. They treated the divergent harmonic series (1 + 1/2 + 1/3 + …) as a finite quantity x and derived the perfect powers sum from it. But that’s not valid — you can make divergent series produce virtually any result you want through such manipulation. A rigorous proof only appeared later.

Why the Result Still Matters

That said, the fact itself remains genuinely striking: the infinite sum of reciprocals of perfect powers converges to exactly 1. No more, no less.

It’s one of those results where the answer feels almost too clean — and that’s precisely what makes number theory worth exploring.