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Topic: The ordering relation on the integers (Read 338 times) |
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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.
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| « Last Edit: Aug 6th, 2015, 7:29pm by 0.999... » |
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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)
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| « Last Edit: Aug 7th, 2015, 12:33pm by towr » |
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Wikipedia, Google, Mathworld, Integer sequence DB
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