Ramanujan's Pi Formula Took 73 Years to Prove

Four constants: 9,801. 1,103. 26,390. 396. On their own, nothing about them signals anything. Together, they produce pi with a speed that still looks, a century later, like a conjuring trick. Take the first term of the series and stop: you already have eight correct digits of pi. Add another: sixteen. Another: twenty-four. Eight new digits, every single term, like someone with very clean handwriting just keeps writing them out for you.

Srinivasa Ramanujan published this formula in 1914. He offered no derivation, no proof, no pathway in. His justification, in full, was that the formula is "extremely rapidly convergent." He checked the first term against the known value, noted that they matched to eight decimal places, and moved on. The single formula that would outlive everything else in that paper was the one he explained least.

For 73 years, nobody could formally prove it worked. And yet a direct descendant of it is, right now, the core algorithm behind record-breaking computations of pi. A working machine. No manual. Decades of people trying to figure out why it never broke.

The number that doesn't belong

A useful place to start is 396, specifically 396 to the fourth power, which is roughly 24.5 billion. Nothing exotic on the surface. But take its natural logarithm, divide by pi, and square the result: you land on 58. A whole number. There is no obvious reason that chain of operations should terminate cleanly.

Run it backward and it gets stranger. Raise e to the power of pi times the square root of 58. Three irrational numbers compounded. The result should unspool into endless decimal noise. Instead, it misses an exact whole number by less than two ten-millionths, and the whole number it nearly lands on is 396 to the fourth, minus 104.

As the STEM in Motion video tracing this mathematical lineage puts it: "On the surface, nothing about e, pi, or the square root of 58 knows anything about 396. Something hidden is forcing them together."

That hidden thing is the actual story.

Gauss, ellipses, and a slow route to circles

The machinery starts with Gauss in 1799, who was not thinking about pi. He was playing with a number-averaging game: take two values, compute their arithmetic mean, compute their geometric mean, feed both results back in, repeat. The two numbers collapse toward each other at extraordinary speed, doubling the number of correct digits on every pass.

Give those two numbers a geometric meaning and it sharpens considerably. An ellipse is defined by two lengths: its major axis and its minor axis. Feed those in as your starting pair. Each averaging step pulls the axes closer together, which means the ellipse gets rounder. Run it to infinity and you have a circle.

What Gauss found is that a specific quantity computed from the ellipse's boundary stays invariant through every averaging step. That means you can compute it from any shape in the sequence, including the circle at the end. And a circle's geometry is built on pi, so the invariant hands you pi almost automatically. Every formula in this lineage is, underneath the notation, just someone finding a clever route back to that circle.

The degree dial

Ramanujan's contribution was to ask a different question about the same objects. Rather than watching one ellipse smooth itself into a circle, he asked: can two different ellipses, one stretched by a whole-number multiple of the other, be connected by an exact equation? Almost always, yes, but the equation is usually monstrous. Nested square roots inside square roots, pages of it. Mathematicians call these modular equations, and the stretching factor is the degree: degree 2, degree 7, degree 58.

At most degree settings, the equations are too ugly to use. But at rare values, everything collapses. The nested roots cancel, and you're left with clean whole-number arithmetic. The video describes it as a dial: "Every setting gives you a different equation, and almost all of them are far too ugly to use."

When a degree collapses, those whole numbers become the coefficients of a series in e to the minus pi times the square root of that degree. Flip the series and the leading whole number sits atop e to the plus pi root of the degree, with only tiny terms trailing behind. That is why 396 to the fourth almost exactly equals e to the pi root 58, plus 104. The near-miss is the collapse, observed from the outside.

More digits per term comes directly from a larger leading denominator, which comes directly from a higher degree. The dial is the only control in the entire system.

The notebooks, the paper, the silence

Ramanujan spent years before 1914 filling private notebooks with collapsed values at dozens of different degrees, computed entirely by hand to fifteen or twenty decimal places. As Cantor's Paradise documents, no published table at the time approached that depth. The notebooks were a private lookup table he had built over years, which is why his formulas looked, to everyone else, like they materialized from nowhere.

His 1914 paper, "Modular Equations and Approximations to Pi," is 23 pages long with 17 formulas for pi or its reciprocal. For most of them, he walks through the method. But near the end, under a section promising "a few series for 1/pi," he lists a family of series whose leading constants increase by a fixed step of 26,390 each time. The final entry is the formula at the top of this piece. The constant 26,390 was never chosen; it is the step size of an arithmetic progression. The 9,801 is simply 99 squared.

He noted the rapid convergence. He did not connect the formula to the modular machinery twenty pages earlier, did not mention degree 58, did not write down a proof. He died in 1920, six years later.

Watson's wall, and what the Borweins had to build

The British mathematician G. N. Watson spent the years after Ramanujan's death working systematically through the notebooks, constructing formal proofs for the modular equations. He made genuine progress. This one series stayed out of reach, and it is worth sitting with what that actually means: Watson was not being careless. He was a careful, thorough mathematician working through material he understood well. The series simply required a general theory of what happens at every degree simultaneously, machinery for turning the dial and knowing precisely what would fall out, and that theory did not yet exist. Checking harder was not the path. You cannot prove what an infinite sum does at term ten million by confirming what it does at term ten.

That theory finally arrived in 1987, when Jonathan and Peter Borwein built enough of the framework to formally prove all 17 of Ramanujan's 1914 formulas. They traced the opening formula to degree 58 and proved, not observed but proved, that the sum equals 1/pi exactly, all the way to infinity, strictly because it is a degree-58 object. Spotting the numerical fingerprint took a glance. Proving it took seven decades and mathematics that had not been invented when Ramanujan wrote the formula down.

Degree 163 and a closed door

Once the degree relationship was understood, it became a systematic method. Higher degree, tighter collapse, more digits per term. David and Gregory Chudnovsky went hunting above degree 58 and stopped at 163, not by choice but because number theory stops them there. The complete list of degrees where the arithmetic comes out perfectly clean contains exactly nine values: 1, 2, 3, 7, 11, 19, 43, 67, and 163. The list is finite and proven to be complete. Degree 58 is not on it; it sits one tier below, which is why it collapses but leaves a residual of two ten-millionths.

Degree 163 misses a whole number by seven ten-trillionths, roughly a quarter of a million times tighter than degree 58. The Chudnovsky formula built on it produces 14 correct digits of pi per term, against the 8 from Ramanujan's. It is the same engine architecture with a denominator ten million times larger: 640,320 cubed in place of 396 to the fourth. The Chudnovsky algorithm, as documented across the mathematical literature, now powers every major pi computation record. The mathematical engine has not changed since 1988.

What he missed on page 4

Here is the part that does not resolve neatly. Early in the 1914 paper, Ramanujan computes e to the power of pi times the square root of 58 by hand, to eight decimal places. He was interested in how close such numbers come to whole integers, a curiosity of a different kind. He records the near-miss. Then, seventeen pages later, in the same paper, in his own handwriting, he writes down the series whose denominator is built on 396 to the fourth, which is precisely the whole number that value nearly lands on.

He never connected them. Both halves of the proof were sitting in his own paper and he did not place them side by side.

The Borweins later showed he had found the degree-58 instance of a theorem that would not formally exist for another 73 years. He was right about it. He just did not know what he had found, or rather, he knew the result without knowing the category the result belonged to.

What do you do with that? Berndt, who spent decades reconstructing Ramanujan's methods from the notebooks, concluded that the notebooks offer no hints about underlying reasoning, and that modern proofs of Ramanujan's results are likely considerably harder than whatever path Ramanujan actually took. That is not a resolution. That is the stalemate restated at a higher level of precision.

Ramanujan was right about a theorem that had no name. He never wrote down the connection that would have explained why. Whether he saw it and considered it too obvious to mention, or whether he genuinely had not put the two pages together, nobody knows. The notebooks are silent. The paper is silent. And the silence is now over a century old.

By Amelia Nwofor, Science Desk Editor