Størmer's theorem

In number theory, Størmer's theorem, named after Carl Størmer, gives a finite bound on the number of consecutive pairs of smooth numbers that exist, for a given degree of smoothness, and provides a method for finding all such pairs using Pell equations. It follows from the Thue–Siegel–Roth theorem that there are only a finite number of pairs of this type, but Størmer gave a procedure for finding them all.

If one chooses a finite set ${\displaystyle P=\{p_{1},p_{2},\dots p_{k}\}}$ of prime numbers then the P-smooth numbers are defined as the set of integers

that can be generated by products of numbers in P. Then Størmer's theorem states that, for every choice of P, there are only finitely many pairs of consecutive P-smooth numbers. Further, it gives a method of finding them all using Pell equations.

Størmer's original procedure involves solving a set of roughly 3^kPell equations, in each one finding only the smallest solution. A simplified version of the procedure, due to D. H. Lehmer, is described below; it solves fewer equations but finds more solutions in each equation.

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