This is an extract from [Apéry's biography](http://peccatte.karefil.com/PhiMathsTextes/Apery.html)
(which some of the people have already enjoyed in
[this answer](https://mathoverflow.net/questions/25630/major-mathematical-advances-past-age-fifty/25631#25631)).



> 
> 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](http://www.oeis.org/A010330) 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]](http://dx.doi.org/10.1215/S0012-7094-77-04433-7).
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](http://dx.doi.org/10.1017/S0305004100032850) indicates
the particular solution
$$
\binom{132}{4}+\binom{190}{4}=\binom{200}{4}
$$
and the trivial infinite family
$$
\binom{2n-1}n+\binom{2n-1}n=\binom{2n}n.
$$



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


