A Prime Factorization Theorem and Number Systems

AI Thread Summary
Andrew Wiles, in his 2016 Abel Prize interview, discusses new number systems where the fundamental theorem of arithmetic, which asserts unique prime factorization, does not apply. An example provided is the ring of integers in the quadratic integer ring \(\mathbb{Z}[\sqrt{-5}]\), where numbers can have multiple irreducible factorizations. This lack of unique factorization complicates mathematical proofs, such as Fermat's Last Theorem. Such number systems challenge traditional arithmetic principles and highlight the complexity of advanced mathematical structures. Understanding these systems is essential for grasping modern number theory concepts.
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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?
 
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e2m2a said:
Summary:: 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?
Could you give us the link? At least to those who are willing to watch almost an hour of content only to guess what somebody else might have possibly meant.

##\mathbb{Z}[\sqrt{-5}]## is a ring without prime factorization, because ##2\cdot 3=(1+\sqrt{-5})\cdot(1-\sqrt{-5})## are two different representations with irreducible factors.
 
fresh_42 said:
##\mathbb{Z}[\sqrt{-5}]## is a ring without prime factorization, because ##2\cdot 3=(1+\sqrt{-5})\cdot(1-\sqrt{-5})## are two different representations with irreducible factors.
If you are not familiar with this notation then note that ##\mathbb{Z}[\sqrt{-5}]## is an example of a quadratic integer ring; specifically it is the 'number system' whose elements ## w ## are defined by two (ordinary) integers ## (a, b) ## such that ## w = a + b \sqrt{-5} ##.

As @fresh_42 states, in this number system the number ## (6, 0) ## is an example of a number that has two distinct irreducible ('prime') factorizations: ## (2, 0) (3, 0) ## and ## (1, 1) (1, -1) ##.
 
fresh_42 said:
Could you give us the link? At least to those who are willing to watch almost an hour of content only to guess what somebody else might have possibly meant.

##\mathbb{Z}[\sqrt{-5}]## is a ring without prime factorization, because ##2\cdot 3=(1+\sqrt{-5})\cdot(1-\sqrt{-5})## are two different representations with irreducible factors.
 
Another one is the a 2×2 matrix that has all the same integer entries. Will not have a fundamental theorem of arithmetic, I think the only unique factorization is a matrix of -1 or 1. There is a paper in the MAA Journal math monthly for more information about matrix number theory.
 
e2m2a said:
Summary:: 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?
The example given by @fresh_42 is exactly what Wiles is talking about. The rings of integers in number fields do not always have unique factorization. If they did Fermat's last thereom would have been much easier to prove.
 
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