/sci/ - Science & Math

>>8043773

Continuing this Rule 2 ad nauseum, particularly fitting formulae to data, give the following rules 3, 4, 5

[math] \displaystyle Rule \;\; 3: \;\;\; \frac{2}{2n+1} = \frac{1}{ \frac{3}{5}(2n+1)} + \frac{1}{3(2n+1)} [/math]

[math] \displaystyle Rule \;\; 4: \;\;\; \frac{2}{2n+1} = \frac{1}{ \frac{4}{7}(2n+1)} + \frac{1}{4(2n+1)} [/math]

[math] \displaystyle Rule \;\; 5: \;\;\; \frac{2}{2n+1} = \frac{1}{ \frac{6}{11}(2n+1)} + \frac{1}{6(2n+1)} [/math]

etc, all of which taken together with obvious patterns coming out above, suggests an /algorithm/ that Ahmose could have used to populated his entire table with fractions of exacty /two/ terms. Whereas what /actually/ happened, in my view, is that Ahmose followed this algorithim /about halfway through/ the problem, then started making changes which I have yet to investigate.

This is all a bit premature, there's something here for me to show by induction, and /I haven't actually double-checked the above yet/, so check it/take it with a grain of salt.

Two possible activities at this point are to build a "revisionist" table, different from what Ahmose actually wrote, where every case is decomposed into exactly two terms, and giving a general statement about the above, closely modeled on "61B". I seem to be retreading the ground mentioned on the wiki page.

Pic related amounts to the algorithm I'm describing, mirroring Eratosthenes' sieve, albeit in the service of a specific problem.