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While reading a mini-bio about Ramanujan, there was an equation that I found somewhat intriguing:$$e^{\pi\sqrt{58}}=24591257751.99999982\ldots\tag1$$ This is interesting in the way that $e^{\pi\sqrt{58}}$ is very close to $24591257752$, and how Ramanujan managed to calculate such a large number back when computers didn't exist! Others include \begin{equation}35\sqrt{\pi}\ln 2=42.9999986\ldots\tag2\\ 51\ln(36\pi)-2e^\pi+\sqrt[4]{21}=196.99999991695\ldots\\ e^{{\frac \pi 4\sqrt{102}}}=800\sqrt3+196\sqrt{51}\\\vdots\end{equation}

And there are much more. So, I'm wondering:

Main Question: Is there some sort of formula, trick or follow-through that you can use that enables you to calculate other fascinating $e$ equations that are very close to an integer?

Note: I'm not the best at mathematics. In fact, I haven't even started Calculus yet. I would love it if the answer also had a detailed description of how you solved this.

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Frank
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  • $e^{\pi \sqrt{163}}$ is even more impressive. See http://math.stackexchange.com/questions/4544/why-is-e-pi-sqrt163-almost-an-integer and https://en.wikipedia.org/wiki/Almost_integer. – lhf Oct 31 '16 at 17:49
  • It's only 19 significant digits. If you have $\pi$ and $\sqrt{58}$ available to that precision, computing the exponential takes only about 20 terms of the power series -- not something you jot down in a minute, but not something you need to spend years on either. – hmakholm left over Monica Oct 31 '16 at 17:49
  • I have edited the question... Is it okay to reopen it now? – Frank Nov 02 '16 at 21:56
  • It is still a duplicate, since the theory behind those formulas is very complicated, and the answers given are enough. See how it mentions Heegner numbers i.e. (complicated) properties of number fields and modular forms, related to the Ramanujan-Sato series – reuns Nov 02 '16 at 22:20
  • @user1952009 Okay then. I give up. But, is there a pdf of 'Sur la théorie des equations modulaires' in english? – Frank Nov 02 '16 at 22:44
  • @Frank The modular forms are not of your level. And before computers were invented, it was common to run such hard computation up to $20$ digits – reuns Nov 02 '16 at 22:56
  • The calculation of $e^{\pi\sqrt{58}}$ is much simpler as Ramanujan did it. The calculation depends on the value of class invariant $$g_{58}=\sqrt{\frac{5+\sqrt{29}}{2}}$$ and the equation $$64(g_{58}^{24}+g_{58}^{-24}) =e^{\pi\sqrt{58}}-24+4372e^{-\pi\sqrt{58}}+\dots$$ All of this is based on the theory of modular equations and presented by Ramanujan in his classic paper Modular Equations and Approximations to $\pi$. The value in your equation $(1)$ appears to be directly taken from the same paper. He gave more examples of such calculations in that paper. – Paramanand Singh May 23 '17 at 06:18

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