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- In combinatorial number theory, the Lambek–Moser theorem splits the natural numbers into two complementary sets using any non-decreasing function and its inverse. It extends Rayleigh's theorem, which splits the natural numbers into complementary sets in a more restricted way by rounding the multiples of two irrational numbers. When a formula is known for the th natural number in a set, the Lambek–Moser theorem can be used to derive a formula for the th number not in the set. There are two parts to the Lambek–Moser theorem. One part states that two 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. As well as the Beatty sequences of Rayleigh's theorem, other pairs of complementary sets of natural numbers to which the Lambek–Moser theorem can be applied include the even numbers and the odd numbers, the evil numbers and the odious numbers, the prime numbers and the non-primes (1 and the composite numbers), and the th powers and non-powers. The Lambek–Moser theorem 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. (en)
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