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Størmer's theorem ：ウィキペディア英語版

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.
==Statement==
Formally, the theorem states that, if one chooses a finite set ''P'' = of prime numbers and considers the set of integers
:

S = \left\\cdots p_k^\mid e_i\in\\right\}

that can be generated by products of numbers in ''P'', then there are only finitely many pairs of consecutive numbers in ''S''. Further, it gives a method of finding them all using Pell equations.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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