Wikipedia:Reference desk/Archives/Mathematics/2021 February 26

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February 26

Prove that any positive integer, can be decomposed to at most norm-equal 2 elements in gauss integer

Hello, I'm trying to prove that any positive integer, can be decomposed to at most norm-equal 2 elements in gauss integer. My thought, write down the number as integer product, and then decompose each prime. My problem that I'm unable to handle gaussian prime numbers such 3. --Exx8 (talk) 22:48, 26 February 2021 (UTC)

I'm confused. First, I assume you mean decompose as a product, in other words factor; you can also decompose things as a sum, so you need to specify. Also, every integer is also a Gaussian integer, so if X is an integer then X taken as the product of 1 factor is a decomposition of X. One is less than two so that would be a decomposition of X into at most two Gaussian integer factors. So did you mean exactly two factors? If so then it can't be done, there is no such factorization for 3. This is a consequence of unique factorization which holds in the Gaussian integers. --RDBury (talk) 02:34, 27 February 2021 (UTC)

Yes as a product. Decomposition to 1 number is not decomposition to 2 numbers.

I meant that if X=YZ, and YZ are equal in their norm, then there is no A,B such X=AB, |A|=|B|,A≠Y, B≠Z up to multiplication by unity.--Exx8 (talk) 06:52, 27 February 2021 (UTC)

I take the claim to be that if $yz=ab$, where the two factors of either product have equal norms, then the factorizations on either side of the equality can be transformed into each other by multiplying these factors by units. Now $5\cdot 5=(4+3i)(4-3i)$ fits the antecedent of the implication, yet there is no unit $u$ such that $4+3i=5u$. The claim holds in the complex numbers: $u=5^{{-}1}(4+3i)$ has norm $1$. But it is not a Gaussian integer, so it is not a unit in the ring of Gaussian integers.  --Lambiam 08:31, 27 February 2021 (UTC)