A binomial generalization of the FLT: Bombieri's Napkin Problem

This is an extract from Apéry's biography (which some of the people have already enjoyed in this answer).

During a mathematician's dinner in Kingston, Canada, in 1979, the conversation turned to Fermat's last theorem, and Enrico Bombieri proposed a problem: to show that the equation $$\binom xn+\binom yn=\binom zn > \qquad\text{where}\quad n\ge 3$$ has no nontrivial solution. Apéry left the table and came back at breakfast with the solution $n = 3$, $x = 10$, $y = > 16$, $z = 17$. Bombieri replied stiffly, "I said nontrivial."

What is the state of art for the equation above? Was it seriously studied?

Edit. I owe the following official name of the problem to Gerry, as well as Alf van der Poorten's (different!) point of view on this story and some useful links on the problem (see Gerry's comments and response). The name is Bombieri's Napkin Problem. As the OEIS link suggests, Bombieri said that

"the equation $\binom xn+\binom yn=\binom zn$ has no trivial solutions for $n\ge 3$"

(the joke being that he said "trivial" rather than "nontrivial"!).

As Gerry indicates in his comments, the special case $n=3$ has a long history started from the 1915 paper [Bökle, Z. Math. Naturwiss. Unterricht 46 (1915), 160]; this is reflected in [A. Bremner, Duke Math. J. 44 (1977) 757--765]. A related link is [F. Beukers, Fifth Conference of the Canadian Number Theory Association, 25--33] for which I could not find an MR link. Leech's paper indicates the particular solution $$\binom{132}{4}+\binom{190}{4}=\binom{200}{4}$$ and the trivial infinite family $$\binom{2n-1}n+\binom{2n-1}n=\binom{2n}n.$$

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Yep, not very similar but at least with a name. :-) –  Wadim Zudilin Jun 7 '10 at 3:51
By the way, this story is one of the reasons I felt that Apéry was stiffed by "mathematical community" (and Bombieri isn't even French). –  Victor Protsak Jun 7 '10 at 3:57
@Wadim, even for $2\binom{x}{n}=\binom{x+2}{n}$ there are infinitely many solutions with arbitrarily large $n$, $n_0=1,n_1=6$ and following $n_{k+1}=6n_k-n_{k-1}$. –  Gjergji Zaimi Jun 7 '10 at 7:17
I guess after removing also the infinite family that comes from $\binom{x}{n}+\binom{x+1}{n}=\binom{x+2}{n}$, one might expect there to be only finitely many solutions. But still there are more "sporadic" solutions out there. FLT is famous for being easy to state and hard to solve. This question seems to be hard to state :-) –  Gjergji Zaimi Jun 7 '10 at 8:15
There is a small-but-crucial difference in the way the story is told in Alf van der Poorten, Notes on Fermat's Last Theorem, page 122: Michel Mendes France reminds me to tell the story of Bombieri's napkin. At the Queen's University number theory meeting in 1979, Roger Apery was a victim of Enrico Bombieri's observation that "the equation $${x\choose n}+{y\choose n}={z\choose n}$$ has no trivial solutions for $n\ge3$." At breakfast, next morning, Apery excitedly reported having spent the night finding the smallest example $x=10$, $y=16$, $z=17$, with $n=3$. Continued, next comment... –  Gerry Myerson Jun 8 '10 at 3:11

Some solutions for $n=3$ can be found at http://www.oeis.org/A010330 where there is also a reference to J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780, MR 19, 837f (but from the review it seems that paper deals with ${x\choose n}+{y\choose n}={z\choose n}+{w\choose n}$).

There are some other solutions at http://www.numericana.com/fame/apery.htm

EDIT Here are some more references for $n=3$:

Andrzej Krawczyk, A certain property of pyramidal numbers, Prace Nauk. Inst. Mat. Fiz. Politechn. Wrocƚaw. Ser. Studia i Materiaƚy No. 3 Teoria grafow (1970), 43--44, MR 51 #3048.

The author proves that for any natural number $m$ there exist distinct natural numbers $x$ and $y$ such that $P_x+P_y=P_{y+m}$ where $P_n=n(n+1)(n+2)/6$. (J. S. Joel)

M. Wunderlich, Certain properties of pyramidal and figurate numbers, Math. Comp. 16 (1962) 482--486, MR 26 #6115.

The author gives a lot of solutions of $x^3+y^3+z^3=x+y+z$ (which is equivalent to the equation we want). In his review, S Chowla claims to have proved the existence of infinitely many non-trivial solutions.

W. Sierpiński, Sur un propriété des nombres tétraédraux, Elem. Math. 17 1962 29--30, MR 24 #A3118.

This contains a proof that there are infinitely many solutions with $n=3$.

A. Oppenheim, On the Diophantine equation $x^3+y^3+z^3=x+y+z$, Proc. Amer. Math. Soc. 17 1966 493--496, MR 32 #5590.

Hugh Maxwell Edgar, Some remarks on the Diophantine equation $x^3+y^3+z^3=x+y+z$, Proc. Amer. Math. Soc. 16 1965 148--153, MR 30 #1094.

A. Oppenheim, On the Diophantine equation $x^3+y^3-z^3=px+py-qz$, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 230-241 1968 33--35, MR 39 #126.

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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. –  Wadim Zudilin Jun 8 '10 at 5:21
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. –  Wadim Zudilin Jun 8 '10 at 6:15
@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. –  Gerry Myerson Jun 8 '10 at 7:35
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. –  Wadim Zudilin Jun 8 '10 at 8:29
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. –  Wadim Zudilin Jun 8 '10 at 8:54

My first instinct is to say it seems unlikely there's been serious progress on this problem for general n. Unlike the Fermat equation, this one is not homogeneous of degree n, which means that it's really a question about points on a surface rather than points on a curve. We don't have a giant toolbox for controlling rational or integral points on surfaces as we do for curves.

In fact, I can't think of any example of a family of surfaces of growing degree where we can prove a theorem like "there are no nontrivial solutions for n > N." OK, I guess one knows this about the symmetric squares of X_1(n) by Merel...

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Re "family of surfaces of growing degree": You have to be careful how you word it. What about $x^{2n}+y^{2n}+z^{2n}=7$? –  Victor Protsak Jun 11 '10 at 17:25

Another paper that mentions the problem is "Explicit Solutions of Pyramidal Diophantine Equations" by L.Bernstein Canad. Math. Bull. Vol. 15(2) from 1972! In fact I realized that this problem could have appeared in literature long before expressed in terms of "figurate numbers". Anyway an interesting list of references (I haven't found most of them yet though) can be found on section D8 of R.Guy's "Unsolved Problems in Number Theory".

Also two more OEIS links with useful information. I would also like to find this article by H. Harborth, "Fermat-like binomial equations", Applications of Fibonacci numbers, Proc. 2nd Int. Conf., San Jose/Ca., August 1986, 1-5 (1988). (Link)

As a conclusion, the problem has been mentioned in several papers, and many special cases have been given a lot of attention. Bombieri doesn't seem to be the original source of the question.

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Thanks, Gjergji, for your digging! Bombieri just gave a name to the problem. :-) How one can get Harborth's article? The link to Guy's unsolved problems is really a must (I always underestimate this book because many of the problems there are of insufficient interest, I can't find there a serious problem for a talented student). –  Wadim Zudilin Jun 8 '10 at 13:58
I'll pass by the library to look for Harborth's article a little later (they apparently have a copy), I couldn't find a link though. I felt the same about Guy's unsolved problems but this is the second time I try to find some information on an open problem and that book turns out to be one of the best reference places. (mathoverflow.net/questions/24265/… ) –  Gjergji Zaimi Jun 8 '10 at 14:10
I've already voted on your answer there. I didn't know that Guy has links to Kurepa's conjecture, but this one is really famous and I studied it from a different prospective (mathoverflow.net/questions/24740/non-real-constants/24752#24752). –  Wadim Zudilin Jun 8 '10 at 14:14
Well it was Kurepa's conjecture with a -1 which didn't seem to be that famous. :-) –  Gjergji Zaimi Jun 8 '10 at 14:24
Yes, I just realized (2nd time!) that 24265 had a wrong question (and it is what is collected in Guy's book :-) ). –  Wadim Zudilin Jun 8 '10 at 14:31