Factoriangular numbers in balancing and Lucas-balancing sequence (Q2210308)

The balancing numbers \(\{B_n\}_{n\ge 0}\) have initial terms \(B_0=0,~B_1=1\) and satisfy the recurrence \(B_{n+2}=6B_{n+1}-B_n\) for all \(n\ge 0\). The Lucas-balancing numbers \(\{C_n\}_{n\ge 0}\) have initial terms \(C_0=1,~C_1=3\) and satisfy the same recurrence relation as the balancing numbers. A factoriangular number is a number of the form \(Ft_n=n!+n(n+1)/2\). In the paper under review the authors study the Diophantine equations \(FT_n=B_m,~C_m\), in nonnegative integer indeterminates \((m,n)\). They show that this equation has no interesting solutions other than the trivial ones for which \(n=0\). The proofs use lower bounds for \(p\)-adic linear forms in logarithms of algebraic numbers and some computations.