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

> 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?