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- What does it mean that there are some number systems where the prime factorization theorem does not hold?

If you go to "The Abel Prize Interview 2016 with Andrew Wiles" on YouTube, there is a statement made by Andrew Wiles beginning at about 4:10 and ending about 4:54 where he mentions there are some new number systems possible where the fundamental theorem of arithmetic does not hold. I don't understand how this is possible. Can someone please explain what he meant by this and what number systems he is talking about?