  This is the English version of the paper: "Complejidad de los n\'umeros
naturales", Gaceta de la Real Sociedad Matem\'atica Espa\~nola 3 (2000)
230--250. In this paper, several conjectures about the complexity of natural
numbers are proposed. In a recent joint paper with H. Altman "Integer
Complexity, stability, and self-similarity" arXiv:2111.00671, we resolve the
last of these conjectures. This is why I consider its translation useful now as
a reference.
