Recently my kids and I ran across the Brahmagupta-Fibonacci identity and noticed that the set consisting of integers that are expressible as the sum of two squares is closed under multiplication. There are other obvious multiplicatively closed subsets of Z such as the powers of a prime, and the ideals of Z.

Question: Is the class of all multiplicative subsemigroups of Z fully characterised, and if so please provide an explanation or reference for us.

  • $\begingroup$ I am not sure if there is a complete characterization. But, we have some more obvious examples such as: set of odd numbers, set of non-multiple of a prime, etc. $\endgroup$ – i707107 Aug 8 '16 at 0:32
  • $\begingroup$ I made some progress (after 2 years :-). If we limit ourselves to sets of positive integers, then prime decomposition allows us to convert this question into: which sets of sequences of non-negative integers are closed under addition (of sequences)? $\endgroup$ – Joseph Johnson Dec 18 '18 at 6:02
  • $\begingroup$ It's great that you have been thinking about this. I forgot about this and forgot even that I commented on this. I do see your progress is very valuable toward completely solving this problem. A related good exercise is "Describe $3\mathbb{N}+5\mathbb{N}$" meaning that "How can we describe an additive closed subset of $\mathbb{N}$ such that it contains $3$ and $5$?" $\endgroup$ – i707107 Dec 20 '18 at 2:16

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