[seqfan] Numbers and 'forms'.
peter.luschny at gmail.com
Fri Jun 27 11:52:59 CEST 2014
(a) Numbers of the form x^2 - 2y^2 with integers x, y.
(b) Numbers represented by the form x^2 - 2y^2.
Note that (a) and (b) mean different things.
0 is an integer and 0 = 0^2 - 2*0^2 therefore 0 is
a member of (a).
0 is not a member of (b) because the form x^2 - 2y^2
does not represent 0 since a form represents n by
integers x, y if and only if x != 0.
Now consider the same game with x^2 + 2y^2.
0 is not a member of http://oeis.org/A035251 because ...?
0 is a member of http://oeis.org/A002479 because ...?
Do (a) and (b) have the same meaning in OEIS?
If 'yes' which one?
So why make so much fuss about this '0'? You can find
the answer for example in Andrew Sutherland's wonderful
'Introduction to Arithmetic Geometry' which is MIT Open
Course Ware. Sutherland writes:
"The constraint that x!=0 is critical, otherwise every
quadratic form would represent 0; the quadratic forms
that represent 0 are of particular interest to us."
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