Lambek-Moser theorem

In combinatorics, Lambek-Moser theorem applies to an increasing arithmetic function with non-negative integral value f(n), Let

f *

be an integral-valued function such that

$f(f^*(n)) < n \le f(f^*(n)+1).$

Then

f * * = f

Let

F (n) = f (n) + n, G = f * (n) + n

Then the result states that $F, G$ are strictly increasing and the ranges of $F, G$ form a partition of the positive integers.

The theorem was discovered by Leo Moser and Joachim Lambek.

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[edit] References

Lambek, J.; Moser, L. (Aug-Sep, 1954). "Inverse and Complementary Sequences of Natural Numbers". The American Mathematical Monthly 61 (7): 454–458. doi:10.2307/2308078.