Binomial coefficients in arithmetic progressions (Q2714315)

The author considers three-term arithmetic progressions of binomial coefficients and polynomial values. The following effective finiteness theorem is proven.NEWLINENEWLINENEWLINETheorem. Let \(n\geq 5\) be an integer and \(m\in \{2,4\}\). Further, let \(f(x)\) be an integer-valued polynomial with \(\deg f(x)\leq n-1,\) and let \(g(x)\in { \mathbb{Z} }[x]\). Then there exists an effectively computable constant \(C\) depending only on \(n\) and the polynomials \(f(x)\) and \(g(x)\) such that if for the integers \(x,y\) with \(x\geq n\), \(y\geq m\) the numbers \(f(x)+g(x), \;{y\choose m}, \;{x\choose n}\) in some order form an arithmetic progression, then \(\max\{x, y\}\leq C\).NEWLINENEWLINENEWLINEThe author also gives all the integer solutions \(x,y\) of the equation \(2{x\choose n}={y\choose m}+k\) in the case when \(0\leq k\leq 10\) and \((n,m)\in \{(2,3),(3,2),(3,4), (4,3), (2,6), (6,2), (4,6), (6,4)\}\).