TITLE: Why is $e^{\pi \sqrt{163}}$ almost an integer?
QUESTION [85 upvotes]: The fact that Ramanujan's Constant $e^{\pi \sqrt{163}}$ is almost an integer ($262 537 412 640 768 743.99999999999925...$) doesn't seem to be a coincidence, but has to do with the $163$ appearing in it. Can you explain why it's almost-but-not-quite an integer in layman's terms (I'm not a mathematician)?

REPLY [2 votes]: I know it's super late but in case anyone has this question in the future, Richard Borcherds made a really digestible video on it in 2020: https://www.youtube.com/watch?v=a9k_QmZbwX8&list=PLar4u0v66vIodqt3KSZPsYyuULD5meoAo&index=85