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In August 1823, Abel wrote a letter to Holmboe with a date:

Copenhague, l’an $ \sqrt[3]{6064321219} $ (en comptant lafraction d´ecimal).

$ 1823 \frac{215}{365} < \sqrt[3]{6064321219} < 1823 \frac{216}{365} $, so the date would be 1823-08-04 (the 216th day, counting the 1st day as 0).

But why did Abel choose such a number? He could have chosen integer $ n $ between

$$ \left( 1823 \frac{215}{365} \right)^3 < 6064303397 \le n \le 6064330729 < \left( 1823 \frac{216}{365} \right)^3 . $$

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  • $\begingroup$ N. H. Abel, "Extraits de quelques lettres a Holmboe." In: L. Sylow and S. Lie (eds.), Œuvres Complètes de Niels Henrik Abel, Vol. 2, Christiania (Oslo): Grøndahl & Søn 1881, pp. 254-255 (Google scan): "Copenhague, l'an $\sqrt[3]{6064321219}$ (en comptant la fraction décimal)" A footnote on p. 254 says this equates to August 3rd, but as the fractional part is $\gt \frac{215}{365}$ it actually equates to 3:39pm on August 4th. $\endgroup$
    – njuffa
    Aug 27, 2023 at 5:15
  • $\begingroup$ The footnote on the book only says *) Le 3 août 1823. but not the time. $\endgroup$
    – puzzlet
    Sep 9, 2023 at 1:36
  • $\begingroup$ I already referenced the foot note in my previous comment: "A footnote on p. 254 says this equates to August 3rd". The footnote is in error, however. Just do your own calculation, no magic is involved. $\endgroup$
    – njuffa
    Sep 9, 2023 at 5:11

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