Talk:Best Approximation from Below to 1 as Sum of Minimal Number of Unit Fractions

From ProofWiki
Jump to navigation Jump to search

Note that:

\(\ds \paren {1}^{-1} + 1\) \(=\) \(\ds 2\)
\(\ds \paren {1 - \frac 1 2}^{-1} + 1\) \(=\) \(\ds 3\)
\(\ds \paren {1 - \frac 1 2 - \frac 1 3}^{-1} + 1\) \(=\) \(\ds 7\)
\(\ds \paren {1 - \frac 1 2 - \frac 1 3 - \frac 1 7}^{-1} + 1\) \(=\) \(\ds 43\)
and, unsurprisingly:
\(\ds \paren {1 - \frac 1 2 - \frac 1 3 - \frac 1 7 - \frac 1 {43} }^{-1}\) \(=\) \(\ds 1806\)

Best approximation as in sum of $4$ unit fractions closest to $1^-$?


Its' infinite version is the greedy Egyptian representation of $1$.

Wells is possibly hinting towards Sylvester's Sequence (is it the same one as the red link on Sylvester's page?):

$2, 3, 7, 43, 1807, 3263443, \dots$

recursively defined by

$a_1 = 2$, $a_n = 1 + \displaystyle \prod_{1 \mathop \le k \mathop < n} a_k$

This sequence is A000058 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

--RandomUndergrad (talk) 14:22, 18 July 2020 (UTC)
Yes, it would be that same "Sylvester's sequence" because I got the list of things named after JJS off his wikipedia page, which is indeed a direct link to that sequence.
So I reckon what we can do with this page is either turn it into Definition:Sylvester's Sequence or a page about Definition:Sylvester's Sequence, or something.
Feel free to make something sensible about this if you have a good idea of how to craft it. --prime mover (talk) 16:10, 18 July 2020 (UTC)