Generalizations and Applications
Dickson used his lemma to prove that, for any given number, there can exist only a finite number of odd perfect numbers that have at most prime factors. However, it remains open whether there exist any odd perfect numbers at all.
The divisibility relation among the P-smooth numbers, natural numbers whose prime factors all belong to the finite set P, gives these numbers the structure of a partially ordered set isomorphic to . Thus, for any set S of P-smooth numbers, there is a finite subset of S such that every element of S is divisible by one of the numbers in this subset. This fact has been used, for instance, to show that there exists an algorithm for classifying the winning and losing moves from the initial position in the game of Sylver coinage, even though the algorithm itself remains unknown.
The tuples in correspond one-for-one with the monomials over a set of variables . Under this correspondence, Dickson's lemma may be seen as a special case of Hilbert's basis theorem stating that every polynomial ideal has a finite basis, for the ideals generated by monomials. Indeed, this restatement of Dickson's lemma was used by Paul Gordan in 1899 as part of a proof of Hilbert's basis theorem.
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