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.
Størmer's original procedure involves solving a set of roughly 3k Pell equations, in each one finding only the smallest solution.
Lehmer's method involves solving the Pell equation for each P-smooth square-free number q other than 2.
Lehmer's paper furthermore shows[3] that applying a similar procedure to the equation where D ranges over all P-smooth square-free numbers other than 1, yields those pairs of P-smooth numbers separated by 2: the smooth pairs are then (x − 1, x + 1), where (x, y) is one of the first max(3, (max(P) + 1) / 2) solutions of that equation.
To find the ten consecutive pairs of {2,3,5}-smooth numbers (in music theory, giving the superparticular ratios for just tuning) let P = {2,3,5}.
Størmer's original result can be used to show that the number of consecutive pairs of integers that are smooth with respect to a set of k primes is at most 3k − 2k.
Lehmer's result produces a tighter bound for sets of small primes: (2k − 1) × max(3,(pk+1)/2).
Størmer's theorem is an important part of his proof, in which he reduces the problem to the solution of 128 Pell equations.
[8] Conrey, Holmstrom & McLaughlin (2013) describe a computational procedure that, empirically, finds many but not all of the consecutive pairs of smooth numbers described by Størmer's theorem, and is much faster than using Pell's equation to find all solutions.