[HN Gopher] New Proofs Probe the Limits of Mathematical Truth
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       New Proofs Probe the Limits of Mathematical Truth
        
       Author : headalgorithm
       Score  : 55 points
       Date   : 2025-02-03 17:34 UTC (5 hours ago)
        
 (HTM) web link (www.quantamagazine.org)
 (TXT) w3m dump (www.quantamagazine.org)
        
       | empath75 wrote:
       | Something that may not be clear when reading this is the
       | distinction between complex numbers and the ring of integers
       | adjoined with _i_.
       | 
       | "Complex numbers" are of the form a+bi where a and b can be any
       | real number -- 1, 5, pi, the square root of 2, -2.14, etc.
       | 
       | The ring of integers adjoined with _i_ are numbers of the form
       | a+bi where a and b are both integers (-1, 5, 34, etc).
       | 
       | You can also, in addition to using _i_ , adjoin any real number
       | to the integers and get a new field with numbers of the form a+bx
       | where a and b are integers and x is any additional number you
       | want to add-- frequently square roots like the square root of
       | two.
       | 
       | This result shows undecidability of diophantine equations in all
       | those fields of integers, but not complex numbers, for which it's
       | easy to prove that there are _always_ solutions.
        
         | Sniffnoy wrote:
         | Note that Z[i] is called the "Gaussian integers" -- you don't
         | have to keep repeating "the ring of integers with i adjoined"!
         | 
         | Also I should point out that in general Z[r] for some r (note
         | obviously r doesn't have to be real!) will contain more than
         | just a+br; it'd only be just a+br if r satisfies a monic
         | quadratic over Z.
         | 
         | (I also have to nitpick and point out that these results apply
         | to _rings_ of integers -- actually more broadly -- but not to
         | the _fields_. Yeah unfortunately mathematicians often abuse the
         | language here, using  "number field" to refer to the ring, and
         | it's annoying. But Hilbert's 10th for Q remains open to my
         | knowledge.)
         | 
         | Edit: Ugh I forgot this site doesn't allow bold
        
       | coldcode wrote:
       | Math is such an interesting field. People can work for decades
       | and not make progress, then discover something in a moment of
       | clarity from some seemingly unrelated problem. As a programmer, I
       | don't have that type of patience.
        
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