Timeline for Generalizing Ramanujan's "1729 story"
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 28, 2016 at 9:23 | comment | added | shane.orourke | From the Scientific American's review of a recent film about Ramanujan: `...why Ramanujan already knew that 1729 was a sum of two cubes in two different ways: he encountered this fact while searching for “near solutions” to the impossible whole number equation $x^3 + y^3 = z^3$. This is revealed at the bottom of this notebook page, which is in Ramanujan’s own handwriting.' And you can see a picture of the relevant part of this notebook in the review. See blogs.scientificamerican.com/guest-blog/… | |
| Apr 27, 2016 at 6:32 | vote | accept | Dominic van der Zypen | ||
| Apr 27, 2016 at 1:43 | answer | added | Joe Silverman | timeline score: 7 | |
| Apr 26, 2016 at 22:26 | answer | added | Lucia | timeline score: 10 | |
| Apr 26, 2016 at 19:41 | comment | added | GH from MO | It is classical that $r_2(k)$ and $r_3(k)$ can be arbitrary large. It is conjectured (but it is also widely open) that for $n\geq 5$ we have $r_n(k)\leq 1$. I don't know the status of $n=4$ from the top off my head. | |
| Apr 26, 2016 at 19:17 | comment | added | Anthony Quas | The way I heard the story described was the Ramanujan saw the integers as his "friends". The fact that he immediately commented that 1729 had this property was not something that occurred to him off the top of his head, but a comment about one of his friends. | |
| Apr 26, 2016 at 19:16 | comment | added | Sylvain JULIEN | Is it that different from finding "off the top of one's head" a representation of a three digit random integer as a sum of four or less squares? | |
| Apr 26, 2016 at 19:12 | comment | added | Dominic van der Zypen | Ok good points -- I guess my astonishment was due to my ignorance (or at least not thinking properly about the issue). | |
| Apr 26, 2016 at 16:03 | comment | added | Todd Trimble | I don't know that it was off the top of his head. There is classical number theory (involving Gaussian primes) about the number of ways of a number can be represented as the sum of two squares, and it wouldn't be unnatural to try to investigate the situation about sums of two cubes, so maybe it's just something he happened to know from past investigations. | |
| Apr 26, 2016 at 15:30 | comment | added | Geoff Robinson | Since Ramanujan was from India, which presumably used Imperial measures at the time, he would probably have been taught at school that there were $1728$ cubic inches in a cubic foot. He would also have known that $729 = 9^{3}$. So it is perhaps not surprising that he was aware that $1729$ was a sum of cubes in two different ways. There also aren't many positive integer cubes less than $1729$. | |
| Apr 26, 2016 at 15:01 | history | asked | Dominic van der Zypen | CC BY-SA 3.0 |