- View SourceHello all and happy New Year,

Could someone tell me, please, if this "stands up"?

All odd numbers of the form (a^2 + b^2) / 2 with both a and b being odd,

that is those numbers that lie equidistant between two odd perfect squares,

are the sum of two other perfect squares, one odd and the other even and of

the form ((a+b)/2)^2 and ((a-b)/2)^2

n = (a^2 + b^2) / 2 = x^2 + y^2

let x = (a+b)/2

let y = (a-b)/2

so that

n = (a^2 + b^2) / 2

can be rewritten as

n = ((x + y)^2 + (x - y)^2) / 2

2n = (x + y)^2 + (x - y)^2

2n = (x + y) * (x + y) + (x - y) * (x - y)

2n = x^2 + xy + xy + y^2 + x^2 -xy -xy + y^2

2n = 2x^2 + 2y^2

n = x^2 + y^2

Cheers,

David Litchfield - View Source--- David Litchfield <Mnemonix@...> wrote:
> Hello all and happy New Year,

(03:22) gp > (a^2+b^2)/2 - (((a+b)/2)^2+((a-b)/2)^2)

> Could someone tell me, please, if this "stands up"?

>

> All odd numbers of the form (a^2 + b^2) / 2 with both a and b being odd,

> that is those numbers that lie equidistant between two odd perfect squares,

> are the sum of two other perfect squares, one odd and the other even and of

> the form ((a+b)/2)^2 and ((a-b)/2)^2

%1 = 0

i.e. if n=(a^2+b^2)/2, then n=(((a+b)/2)^2+((a-b)/2)^2)

Phil

=====

The answer to life's mystery is simple and direct:

Sex and death. -- Ian 'Lemmy' Kilminster

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http://mailplus.yahoo.com - View Source--- David Litchfield <mnemonix@...> wrote:
> Cheers for the reply, Phil. Could you explain one thing:

It means I'm running The PARI Group's "gp" package at 3:22am.

> > (03:22) gp >

> I'm not sure exactly what this means.

It's free, it's incredibly powerful, and it answers these kinds of questions

without even breaking a sweat.

> Further, I take it all I had need to have done was

I was simply rewording the identity

>

> > i.e. if n=(a^2+b^2)/2, then n=(((a+b)/2)^2+((a-b)/2)^2)

(a^2+b^2)/2-(((a+b)/2)^2+((a-b)/2)^2) = 0

into a form more similar to your original question.

Phil

=====

The answer to life's mystery is simple and direct:

Sex and death. -- Ian 'Lemmy' Kilminster

__________________________________________________

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http://mailplus.yahoo.com - View SourceSo we can write:

(a+b)/2)^2+((a-b)/2)^2

=a^2/4 + ab/2 + b^2/4 + a^2/4 - ab/2 + b^2/4

=a^2/2 + b^2/2

Jon Perry

perry@...

http://www.users.globalnet.co.uk/~perry/maths/

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