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It's well-known that $ e^{\pi \sqrt{d}} $ is almost an integer when $ d $ is taken to be one of the large Heegner numbers $ d = 43, 67, 163 $.

I'm interested to know what the history of this discovery was like. Specifically, did the "empirical discovery" of the almost-integer values of $ e^{\pi \sqrt d} $ come before or after the theory of complex multiplication was already well understood? In other words, was the discovery that $ e^{\pi \sqrt d} $ was almost an integer for some small values of $ d $ accidental or was it motivated by known properties of the $ j $-invariant?

Ege Erdil
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1 Answers1

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On p. 138 of Ranjan Roy's Elliptic and Modular Functions from Gauss to Dedekind to Hecke, he writes that Hermite gave the $q$-expansion of the $j$-function (not using modern notation) and from this estimated $e^{\pi\sqrt{43}}$ and $e^{\pi\sqrt{163}}$ as being very close to integers.

KCd
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  • I think there was some discussion of this in the linked question as well, but are we confident that Hermite knew $ j((1 + \sqrt{-163})/2) $ should be an integer because $ 163 $ is a Heegner number? If so, I think the important question is whether Hermite was the first person to notice almost-integers of the form $ e^{\pi \sqrt d} $. – Ege Erdil Jun 05 '23 at 05:08
  • See the bottom of p. 48 of Hermite's book : https://gdz.sub.uni-goettingen.de/id/PPN598770909?tify=%7B%22pages%22%3A%5B56%5D%2C%22pan%22%3A%7B%22x%22%3A0.476%2C%22y%22%3A0.608%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.798%7D. He refers there to $e^{\pi\sqrt{\Delta}}$ for suitable integers $\Delta$ as "transcendental", but its transcendence was not proved until the Gelfond-Schneider theorem in 1934 (view $e^{\pi\sqrt{\Delta}}$ as a value of $i^{i\sqrt{\Delta}} = a^b$ for algebraic $a = i$ and $b = i\sqrt{\Delta}$). – KCd Jun 05 '23 at 06:14