Størmer's theorem states that for any fixed finite set of primes, there are only finitely many pairs of consecutive positive integers whose prime factors all belong to . Equivalently, the equation has only finitely many solutions in positive -smooth integers.
For , the pairs include and . 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.
Articles by others on the same topic
Størmer's theorem is a result in number theory that pertains to the distribution of prime numbers. Specifically, it provides conditions under which certain integer sequences can have a density of primes. More precisely, Størmer's theorem can be described in the context of sequences of integers defined by a linear recurrence relation.