In 1985, Gosper used the not-yet-proven formula by Ramanujan
$$\frac{ 1 }{\pi } = \frac{2\sqrt{2}}{99^2}\cdot \sum_{n=0}^\infty \frac{(4n)!}{(n!)^4}\cdot\frac{26390 n+1103}{99^{4n}}$$
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?
Best Answer
What follows is taken directly from Borweins' Pi and the AGM.
Let $N$ be a positive number and $q_N=e^{-\pi\sqrt{N}}$ and $$k_N=\frac{\vartheta_{2}^{2}(q_N)}{\vartheta_{3}^{2}(q_N)},k'_N=\sqrt{1-k_N^2},G_N=(2k_Nk'_N)^{-1/12},g_N=\left(\frac{2k_N}{{k'} _N^{2}}\right)^{-1/12}\tag{1}$$ where $\vartheta _2,\vartheta_3$ are theta functions of Jacobi defined by $$\vartheta_{2}(q)=\sum_{n\in\mathbb {Z}} q^{(n+(1/2))^2},\, \vartheta_{3}(q)=\sum_{n\in\mathbb {Z}} q^{n^2}\tag{2}$$ Borwein brothers define another variable $$\alpha(N) =\frac{E(k'_N)} {K(k_N)} - \frac{\pi} {4K^2(k_N)}\tag{3}$$ where $K, E$ denote standard elliptic integrals $$K(k) =\int_{0}^{\pi/2}\frac{dx}{\sqrt{1-k^2\sin^2x}},\,E(k)=\int_{0}^{\pi/2}\sqrt{1-k^2\sin^2x}\,dx\tag{4}$$ It is well known that $k_N, k'_N, \alpha(N) $ are algebraic when $N$ is a positive rational number. We assume $N$ to be a positive integer unless otherwise stated.
Borweins present two class of series for $1/\pi$ based on Ramanujan's ideas which are of interest here: $$\frac{1}{\pi}=\sum_{n=0}^{\infty} \dfrac{\left(\dfrac{1}{4}\right)_n\left(\dfrac{2}{4}\right)_n\left(\dfrac{3}{4}\right)_n} {(n!)^{3}}d_n(N) x_N^{2n+1}\tag{5}$$ where $$x_N=\frac{2} {g_N^{12}+g_N^{-12}} =\frac{4k_N{k'} _{N}^{2}}{(1+k_N^2)^2}, \\ d_n(N) =\left(\frac{\alpha(N) x_N^{-1}}{1+k_N^2}+\frac{\sqrt{N} }{4}g_N^{-12}\right) +n\sqrt{N} \left(\frac{g_N^{12}-g_N^{-12}}{2}\right) \tag{6}$$ and $$\frac{1}{\pi}=\sum_{n=0}^{\infty} (-1)^n\dfrac{\left(\dfrac{1}{4}\right)_n\left(\dfrac{2}{4}\right)_n\left(\dfrac{3}{4}\right)_n} {(n!)^{3}}e_n(N) y_N^{2n+1} \tag{7}$$ where $$y_N=\frac{2} {G_N^{12}-G_N^{-12}} =\frac{4k_Nk'_{N}}{1-(2k_Nk'_N)^2}, \\ e_n(N) =\left(\frac{\alpha(N) y_N^{-1}}{{k'} _{N} ^{2}-k_N^2}+\frac{\sqrt{N} }{2}k_N^2G_N^{12}\right) +n\sqrt{N} \left(\frac{G_N^{12}+G_N^{-12}}{2}\right)\tag{8}$$ The series in question is based on $(5)$ with $N=58$. Another (not so famous but equally remarkable) series given by Ramanujan using $(7)$ with $N=37$ is as follows: $$\frac{4}{\pi}=\frac{1123}{882}-\frac{22583}{882^3}\cdot\frac{1}{2}\cdot\frac{1\cdot 3}{4^2}+\frac{44043} {882^5}\cdot\frac{1\cdot 3}{2\cdot 4}\cdot\frac{1\cdot 3\cdot 5\cdot 7}{4^2\cdot 8^2}-\dots\tag{9}$$ Borweins mention that the values of $\alpha(37),\alpha(58)$ (leading to $1123$ in series $(9)$ and $1103$ in series in question) were obtained by calculating $e_0(37)$ and $d_0(58)$ to high precision.
The details of these calculations are not revealed by Borwein Brothers. But it appears that using value of $\pi$ given by Gosper and the series in question (as well as series $(9)$) one can get the values of $d_0(58),e_0(37)$ to high precision. Further some amount of computation is needed to get the minimal polynomials for $\alpha(37),\alpha(58)$. And then one can get their values in closed form as radicals.
The procedure is similar to what you have done in one of your papers dealing with evaluation of coefficients in Chudnovsky formula but the analysis is probably more complicated because the functions involved here are not like the $j$ invariant taking integer values.
To sum up, Gosper role was important in this proof on two fronts. First is the computation itself and second is that his computation brought Ramanujan's formula into limelight. It was sitting there in his paper Modular equations and approximations to $\pi$ since 1914 and no one before Gosper even looked at it.
Also as most references indicate Gosper's computation was done earlier without any knowledge of Borwein's work which got published later. So Gosper was not aware of the proof of the formula and didn't know that his computation would someday be used as a part of the proof of the series which he computed.