jmc

We rewrite the given recursion as $$a_k{a}_{k+1}=a_{k-1}{a}_k-3.$$This implies that the numbers $a_0{a}_1,a_1{a}_2,a_2{a}_3,\dots$ form an arithmetic sequence with common difference $-3$. We have $a_0{a}_1=37\cdot 72$ and $a_{m-1}{a}_m=0$ (because $a_m=0$). Since those two terms are $m-1$ terms apart, we have $$a_{m-1}{a}_m-a_0{a}_1=0-37\cdot 72=-3(m-1),$$so $$m=37\cdot 24+1=\boxed{889}.$$