# Binomial triplets

Solutions to the equation $$\dbinom{a}{n}+\dbinom{b}{n}=\dbinom{c}{n}$$ I will refer to as 'Binomial triplets of order $n$'. These triplets describe simplicial $n$-polytopic numbers that can be written as the sum of two $n$-polytopic numbers.

The above equation has infinitely many solutions for $n=1$ (trivial) and $n=2$ (the triplets $(a,b,c) =$ $(3,3,4)$, $(4,6,7)$, $(6,7,9)$, $(5,10,11)$, $(7,10,12)$, etc.). Also, regardless the value of $n$, it allows for at least one solution $(2n-1,2n-1,2n)$.

Is it true that for any $n$ the above equations has infinitely many solutions? Are there infinitely many $n$th order Binomial triplets?

It appears this problem is known as Bombieri's Napkin Problem.[\Edit]

• For $n=3$, this is $f(a)+f(b)=f(c)$ for some particular cubic polynomial $f$. I'd expect such an equation to have only finitely many solutions, and ditto for any $n>3$. Note $${10\choose3}+{16\choose3}={17\choose3}$$ – Gerry Myerson Aug 16 '14 at 12:50
• Two more: $\binom{22}{3}+\binom{56}{3}=\binom{57}{3}$ and $\binom{32}{3}+\binom{57}{3}=\binom{60}{3}$. – Johannes Aug 17 '14 at 14:15
• OEIS has listed quite a few tetrahedral numbers that are the sum of two other tetrahedral numbers: oeis.org/A034404 . – Johannes Aug 17 '14 at 14:41
• So, I knew a lot more about this problem 4 years ago than I knew yesterday. Anyway, it seems that what is known about the question is at the MO link. – Gerry Myerson Aug 18 '14 at 1:40

Some computations were reported by John Leech, Some solutions of Diophantine equations, Math Proc Camb Phil Soc 53 (1957) 778-780. For $n=3$, he reported 17 solutions with $c<500$. For $n=4$ and $c<500$ he found only $(a,b,c)=(190,132,200)$ (and the trivial $(7,7,8)$). For $n\ge5$, he found only the trivial $(2n-1,2n-1,2n)$ (it's not clear to me what range of values of $n$ and of $c$ were tested).
Some other solutions are reported at http://www.numericana.com/fame/apery.htm One infinite family begins $(n,a,b,c)=(6,19,19,21),(35,118,118,120),(204,695,695,697),\dots$. Another begins $(n,a,b,c)=(6,14,15,16),(40,103,104,105),(273,713,714,715),\dots$. This at least shows there are nontrivial solutions for arbitrarily large $n$.