= Størmer's theorem
{c}
<Størmer's theorem> states that for any fixed finite <set> $S$ of <primes>, there are only finitely many pairs of consecutive positive <integers> whose <prime factors> all belong to $S$. Equivalently, the equation $y-x=1$ has only finitely many solutions in positive $S$-smooth <integers>.
For $S=\{2,3\}$, the pairs include $(1,2),(2,3),(3,4)$ and $(8,9)$. The underlying method reduces the search to finitely many <Pell equations> by separating square factors from the products of allowed primes. The allowed prime set must stay fixed; allowing it to grow with the candidate <integers> would remove the finiteness assertion.
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