6 ms·
Ramanujan’s observation about 1729 is actually quite interesting and not at all whimsical or silly. It was that is is the smallest number expressible as the sum
by falsedan 9y ago
Ramanujan’s observation about 1729 is actually quite interesting and not at all whimsical or silly. It was that is is the smallest number expressible as the sum of two cubes in two different ways:
1^3 + 12^3
10^3 + 9^3
Ramanujan probably derived this on the spot (by familiarity with the low cubic powers, and eliminating all the other sums of the low cubics [(n-2)^3 + (n-1)^3 only exceeds n^3 for n>7]). But he may have already been familiar with this property, since it was first found by Frenicle (along with 4 other, higher numbers expressible as such a pair of sums of cubes) as a solution to a challenge put forth by Euler.
- falsedan 9y agoPlease don’t steal my bank cards.
- DonbunEf7 9y agoRamanujan may have been aware of 1728 == 12² because of 12's recurrence within number theory. (Well, really, 24's recurrence. [1]) He wrote down several equations [2] which feature 12 and 24, and given the time in which he lived, he may have been acquainted with the work of Felix Klein and others [3] as well. The other sum is 1000 + 729, much easier for a decimal-preferring mentat like Ramanujan to pull from thin air. [1] http://math.ucr.edu/home/baez/numbers/ http://math.ucr.edu/home/baez/numbers/ [2] https://en.wikipedia.org/wiki/Mock_modular_form https://en.wikipedia.org/wiki/Mock_modular_form [3] https://en.wikipedia.org/wiki/J-invariant https://en.wikipedia.org/wiki/J-invariant
- falsedan 9y ago1728 = 12^3
- srtjstjsj 9y agoThat's not whimsical and silly?
- falsedan 9y agoI don’t think so. I think it’s quite interesting! Especially trying to understand Ramanujan’s thought process that produced this insight!