[seqfan] Re: Numbers and 'forms'.

[Neil Sloane]

Peter said " a form represents n by
integers x, y if and only if x != 0."

The authorities disagree.

 David Cox, "Primes of the Form x^2+ny^2" (Wiley), first edition
page 24; second edition, p 22; says
"An integer m is represented by a form f(x,y) if the equation
m = f(x,y) has an integer solution in x and y."

Buell, Binary Quadratic Forms, Springer, page 1,
says
A form f(x,y) represents an integer m, positive, negative, or zero, if
there exist
integers x_0, y_0 such that  a x_0^2+b x_0 y_0 + c y_0^2 = m."




On Fri, Jun 27, 2014 at 5:52 AM, Peter Luschny <peter.luschny at gmail.com>
wrote:

> Please consider:
>
> (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?
>
> Confused,
> Peter
>
> P.S.
> 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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>



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