Timeline for A binomial generalization of the FLT: Bombieri's Napkin Problem
Current License: CC BY-SA 3.0
14 events
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| S Oct 29, 2013 at 11:34 | history | suggested | Abhimanyu Pallavi Sudhir | CC BY-SA 3.0 |
Select text,p1.id,name from posthistory p inner join posthistorytypes p1 on p.posthistorytypeid=p1.id where postid=80978
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| Oct 29, 2013 at 11:30 | review | Suggested edits | |||
| S Oct 29, 2013 at 11:34 | |||||
| Jun 10, 2010 at 2:29 | vote | accept | Wadim Zudilin | ||
| Jun 10, 2010 at 2:29 | comment | added | Wadim Zudilin | Thanks, Gerry! So, D8 is about $n=3$ as well. I would say that it's a right time to accept your answer, even both your and Gjergji's responses and comments gave me a real understanding of what is known and what is not. The problem for $n>3$, besides the "trivial" infinite families, is pretty open... | |
| Jun 10, 2010 at 0:11 | comment | added | Gerry Myerson | D8: "Wunderlich asks for (a parametric representation of) all solutions of the equation $x^3+y^3+z^3=x+y+z$. Bernstein, S. Chowla, Edgar, Fraenkel, Oppenheim, Segal, and Sierpinski have given solutions, some of them parametric, so there are certainly infinitely many. Eighty-eight of them have unknowns less than 13000. Bremner has effectively determined all parametric solutions." Guy then gives 10 references, most of which have already been given here, and none of them more recent than the Bremner paper from 1977. | |
| Jun 9, 2010 at 12:01 | comment | added | Wadim Zudilin | That's a serious update! Somebody borrowed Guy's Unsolved Problems from the library, so that I can't check D8 for the moment... Another unreachable text is Bökle's from 1915 which as far as I understand from Bremner's paper already has infinitely many solutions for $n=3$. | |
| Jun 9, 2010 at 0:51 | history | edited | Gerry Myerson | CC BY-SA 2.5 |
Added several references and some commentary
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| Jun 8, 2010 at 13:36 | comment | added | Wadim Zudilin | Yes: F. Beukers, Integral points on cubic surfaces, CRM Proceedings and Lecture Notes 19 (1999), 25-33. My google search produced a different list but then I added "Beukers". :-) The paper does not provide any additional information for the cubic case: "A systematic account of these polynomial solutions is given by A. Bremner." | |
| Jun 8, 2010 at 8:54 | comment | added | Wadim Zudilin | It seems that Frits' PhD was more about the Ramanujan--Nagell equation (MR0541444), and his interests in the binomial FLT are reflected in that Canadian publication only. | |
| Jun 8, 2010 at 8:29 | comment | added | Wadim Zudilin | O-oh! I now understand Frits' explanation of why he was tied to Apery in his original research: his PhD thesis is reflected in the reference you provide (it's not easy to get it here but I'll do) and later he gave the most elegant proof of the irrationality of $\zeta(2)$ and $\zeta(3)$. There are so many nice contemporary stories in maths... Thank you very much for these links to Beukers and Bremner! I'll probably need to ask some details directly from Frits. | |
| Jun 8, 2010 at 7:35 | comment | added | Gerry Myerson | @Wadim, thanks for the link to Leech. He gives one solution for $n=4$. Another source for $n=3$ is Frits Beukers, Integral points on cubic surfaces, Fifth Conference of the Canadian Number Theory Association, 25-33 - see mid-page 26, and in particular Beukers' reference to Andrew Bremner, Integer points on a special cubic surface, Duke Math J 44 (1977) 757-765. Beukers traces $n=3$ back to 1915. | |
| Jun 8, 2010 at 6:15 | comment | added | Wadim Zudilin | Gerry, I personally don't like the numericana link. Leech, among some things, briefly discusses $n=3$ (dx.doi.org/10.1017/S0305004100032850), so all the links are in fact about $n=3$, except more general results on numericana which are discussed in the above comments by Gjergji. Are the solutions for $n=3$ "spontaneous"? Are there examples for $n>3$, different from the two Gjergji's families? I am really happy of getting the name and references, but the question on was a serious research done towards this problem remains. | |
| Jun 8, 2010 at 5:21 | comment | added | Wadim Zudilin | Gerry, thank you for the links! I vote. BTW, this will let you edit my post (>2000 reps) by adding your nice comments from Alf. Keep in mind that the biography above represents Apery's original point of view, while Alf sees everything from his side. | |
| Jun 8, 2010 at 4:12 | history | answered | Gerry Myerson | CC BY-SA 2.5 |