There are 7 equations between binomial coefficients that aren't explained by any of the four known patterns. People have checked all binomial coefficients less than 10⁶⁰, and these 7 coincidences are all they've found! It's conjectured that these "magnificent seven" are the only ones.
Yesterday, on Mastodon, Paul Schwahn found a beautiful explanation of the first of these coincidences. I tried using similar techniques to explain others, but failed. However, I made two amusing little discoveries in the process!
Anyway, six coincidences remain unexplained! As far as I know, anyway.
(Of course the whole concept of "explaining a coincidence" is a fuzzy business: as we allow more and more complicated "explanations" we're eventually sure to find one - but the more complicated it is, the less it feels like an "explanation". This fact is obvious, hence not very interesting.)
https://johncarlosbaez.wordpress.com/2026/09/20/binomial-coefficient-coincidences/
John Carlos Baez
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2026-09-21 08:47:42 UTC
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2026-09-21 08:47:42 UTCEvent JSON
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"content": "There are 7 equations between binomial coefficients that aren't explained by any of the four known patterns. People have checked all binomial coefficients less than 10⁶⁰, and these 7 coincidences are all they've found! It's conjectured that these \"magnificent seven\" are the only ones.\n\nYesterday, on Mastodon, Paul Schwahn found a beautiful explanation of the first of these coincidences. I tried using similar techniques to explain others, but failed. However, I made two amusing little discoveries in the process!\n\nAnyway, six coincidences remain unexplained! As far as I know, anyway.\n\n(Of course the whole concept of \"explaining a coincidence\" is a fuzzy business: as we allow more and more complicated \"explanations\" we're eventually sure to find one - but the more complicated it is, the less it feels like an \"explanation\". This fact is obvious, hence not very interesting.)\n\nhttps://johncarlosbaez.wordpress.com/2026/09/20/binomial-coefficient-coincidences/",
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