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I found this formula attributed to Kronecker relating solutions of Pell equation to exponential sum:

$$ 4 + \sqrt{17} \approx \frac{2}{9} e^{(5/18) \pi \sqrt{17}} \text{ and } \frac{1}{\sqrt{5}}e^{(1/10) \pi \sqrt{85}}$$

I can run this through a calculator and find both estimates quite convincing:

  • $4 + \sqrt{17} = 8.12310\dots$

  • $ \frac{2}{9} e^{(5/18) \pi \sqrt{17}} = 8.117409\dots $

  • $ \frac{1}{\sqrt{5}}e^{(1/10) \pi \sqrt{85}} = 8.0985607\dots $

These are impressive but not out-of-this-world. They are related to pell equation $t^2 - 17u^2 = - 1$ supposedly. And we are supposed to estimate: $$ \log \Big(t + u \sqrt{17}\Big)$$

in terms of theta functions. Can any one supply the details here. I know similar problems involving $e^{\pi\sqrt{163}}$ that are related to the class number formula.

What is the name for the link to these approximations and Pell equation?

See also History of the theory of numbers, Vol 2 (Eugene Dickson) p 378

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    $\begingroup$ These do not look remotely impressive to me! Can you give any more details? The exp(pi.sqrt(163)) story is to do with modular forms and imaginary quadratic fields; for real quadratic fields the story is different. The log of the fundamental solution to Pell's equation will show up in the formula of a special value of an L-function by the class number formula (so it's possible to imagine that $4+sqrt(17)$ is close to exp of something explicit) but I don't know how to go further. $\endgroup$
    – znt
    Commented Dec 12, 2016 at 11:25
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    $\begingroup$ The name is Kronecker's limit formula. Kronecker's solution of Pell's equation by means of special values of Dedekind's eta function was obtained by the limit formula (Grenzformel), and was regarded at the time as a crowning jewel of the theory of modular functions. You will enjoy reading about this in the last chapter of Andre Weil's book (Elliptic functions according to Eisenstein and Kronecker), or in Vladut's book on the Jugendtraum. The limit formula, giving close approximations of this type, was at the basis of the Gelfond-Linnik-Baker solution to Gauss's class number one problem. $\endgroup$
    – Vesselin Dimitrov
    Commented Dec 12, 2016 at 16:20

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