This is a copy of my question on MSE (https://math.stackexchange.com/questions/3372432) because this forum seems better suited for historical questions:

In 1985, Gosper used the not-yet-proven formula by Ramanujan

$$\frac{ 1 }{\pi } = \frac{2\sqrt{2}}{99^2}\cdot \sum_{n=0}^\infty \frac{(4n)!}{(n!)^4}\cdot\frac{26390 n+1103}{396^{4n}}$$

to compute $17\cdot10^6$ digits of $\pi$, at that time a new world record.

Here (https://www.cs.princeton.edu/courses/archive/fall98/cs126/refs/pi-ref.txt) it reads:

There were a few interesting things about Gosper's computation. First, when he decided to use that particular formula, there was no proof that it actually converged to pi! Ramanujan never gave the math behind his work, and the Borweins had not yet been able to prove it, because there was some very heavy math that needed to be worked through. It appears that Ramanujan simply observed the equations were converging to the 1103 in the formula, and then assumed it must actually be 1103. (Ramanujan was not known for rigor in his math, or for providing any proofs or intermediate math in his formulas.) The math of the Borwein's proof was such that after he had computed 10 million digits, and verified them against a known calculation, his computation became part of the proof. Basically it was like, if you have two integers differing by less than one, then they have to be the same integer.

Now my historical question: Who was the first to prove this formula? Was it Gosper because he added the last piece of the proof, or was it the Borweins, afterwards? And was Gosper aware of this proof when he did his computation?

  • 5
    $\begingroup$ According to the paper Ramanujan's Series for $\frac{1}{\pi}$, Gosper did not prove the result. Refering to J. M. Borwein and P. B. Borwein, "Pi and the AGM; A Study in Analytic Number Theory and Computational Complexity, Wiley, New York, 1987", "In 1987, Jonathan and Peter Borwein succeeded in proving all 17 of Ramanujan's series for $\frac{1}{\pi}$." $\endgroup$
    – nwr
    Dec 18, 2019 at 18:50

1 Answer 1


I asked Bill Gosper, and here is is response:

There was email on this at the time, which I may be able to dig up. But, as I recall, when I began my computation, the Borwein brothers had proved that if Ramanujan's formula did not equal π, it differed from π by at least 10^-3000000, so that as my computation passed the 3000000 mark in agreement with Kanada's 16000000 digit AGM computation, it served to complete the Borwein proof. But by the time my computation reached 17000000, the Borwein's resolved their ambiguity without my empirical confirmation. Their completed proof is almost certainly in their Pi and the AGM book. Tito Piezas and the Chudnovsky brothers have presumably put the matter to rest with their climactic series based on √163, the final Heegner number.

Just to clarify, I did not crank out π as a decimal string. I resumably summed Ramanujan's series as an exact rational number on a Symbolics computer with unlimited integers, converting to decimal now and then for comparison with Kanada, but with the ultimate purpose of computing the {3,7,15,1,292,...} continued fraction, which is actually mathematically interesting, as opposed to useless decimal or binary, which is actually an encryption. Regrettably, (almost) everybody ignored my continued fraction and wasted their time computing (eventually trillions of) useless digits.


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