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        riddles >> medium >> The ordering relation on the integers
        
(Message started by: 0.999... on Aug 6^th, 2015, 7:28pm)

Title: The ordering relation on the integers
Post by 0.999... on Aug 6^th, 2015, 7:28pm

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, [hide]x < y if and only if x does not equal y and there is r such that y=x+r^2[/hide].

Title: Re: The ordering relation on the integers
Post by towr on Aug 7^th, 2015, 12:32pm

[hide]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)[/hide]