#16308: Improve sums of squares
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Reporter: jdemeyer | Owner:
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.3
Component: number theory | Resolution:
Keywords: | Merged in:
Authors: Jeroen Demeyer | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/jdemeyer/ticket/16308 | 6b94a6576031e6e341143a4fc1fd83263a800ed9
Dependencies: | Stopgaps:
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Comment (by pbruin):
Replying to [comment:18 jdemeyer]:
> Replying to [comment:14 leif]:
> > I don't think anybody would expect e.g. `two_squares(25)` to return
the trivial solution `(0,5)` instead of `(3,4)`.
>
> From a number-theory point of view, there is nothing wrong with `(0,5)`.
In fact, in some cases, the only solution is the "trivial" one, for
example for 9.
I agree. Given that 0 is a square, it makes complete sense to allow it as
a term in a sum of squares. That convention makes a lot of things nicer,
for example:
- every square is also a sum of two squares, and more generally every sum
of ''k'' squares is also a sum of ''k'' + 1 squares;
- ''n'' is a sum of 2 squares if and only if ''n'' can be factored non-
trivially in '''Z'''[''i''];
- ''n'' is a sum of ''k'' squares if and only if the lattice
'''Z'''^''k''^ has a point of squared distance ''n'' from the origin.
There are also nice formulae for the number of ways to write ''n'' as a
sum of ''k'' squares if you interpret this number as the number of points
(''x'',,1,, , ... , ''x'',,k,,) in '''Z'''^''k''^ with ''x'',,1,,^2^ + ...
+ ''x,,k,,''^2^ = ''n''.
--
Ticket URL: <http://trac.sagemath.org/ticket/16308#comment:22>
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