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   The ordering relation on the integers
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   Author  Topic: The ordering relation on the integers  (Read 328 times)
0.999...
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The ordering relation on the integers  
« on: Aug 6th, 2015, 7:28pm »
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Find a way to express the (usual) ordering x<y on the integers, by referencing only the operations + and * (and the relation =).  
 
As an example of what I am looking for: doing this over the real numbers is easy, x < y  if and only if x does not equal y and there is r such that y=x+r^2.
« Last Edit: Aug 6th, 2015, 7:29pm by 0.999... » IP Logged
towr
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Re: The ordering relation on the integers  
« Reply #1 on: Aug 7th, 2015, 12:32pm »
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x < y  if and only if x != y and there are r,s,t,u such that y = x + r2 + s2 + t2 + u2  (using the theorem that every non-negative integer can be expressed as the sum of 4 squares)
« Last Edit: Aug 7th, 2015, 12:33pm by towr » IP Logged

Wikipedia, Google, Mathworld, Integer sequence DB
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