The solutions of your equation can be injected to the solutions of the $S$-unit equation over $\mathbb{Q}$, where $S=\{\infty,2,3,5,p\}$. As the latter is known to have finitely many solutions by the results of Siegel, Mahler, Lang (see Ch. 5 in Bombieri-Lang: Heights in Diophantine geometry), your equation also has finitely many solutions.
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