A recent conjecture is proved asserting that \(2, 5\), and \(34\) are the only Fibonacci numbers that are of the form \(n! + n(n+1)/2\) for some integer \(n\). The main tool to prove it is an upper bound for a non-zero \(p\)-adic linear form in two logarithms of algebraic numbers. Computer checking (some brief and some extensive) is indispensable for the proof.