Lambek–Moser theorem

The Lambek–Moser theorem is a mathematical description of partitions of the natural numbers into two complementary sets. For instance, it applies to the partition of numbers into even and odd, or into prime and non-prime (one and the composite numbers). There are two parts to the Lambek–Moser theorem. One part states that any two non-decreasing integer functions that are inverse, in a certain sense, can be used to split the natural numbers into two complementary subsets, and the other part states that every complementary partition can be constructed in this way. When a formula is known for the ${\displaystyle n}$th natural number in a set, the Lambek–Moser theorem can be used to obtain a formula for the ${\displaystyle n}$th number not in the set.

The Lambek–Moser theorem belongs to combinatorial number theory. It is named for Joachim Lambek and Leo Moser, who published it in 1954, and should be distinguished from an unrelated theorem of Lambek and Moser, later strengthened by Wild, on the number of primitive Pythagorean triples. It extends Rayleigh's theorem, which describes complementary pairs of Beatty sequences, the sequences of rounded multiples of irrational numbers.