Let be square free. Consider .

For which conditions on is ?

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- May 9th 2010, 08:19 AMEinStoneQuadratic Number Field
Let be square free. Consider .

For which conditions on is ? - May 9th 2010, 09:13 AMDinkydoe
With the abc-formula we have

Hence we must have for some square

( ) - May 9th 2010, 09:51 AMEinStone
Could you enlarge on that, what are the conditions on a,b,c?

- May 9th 2010, 10:26 AMDinkydoe
Ok.

So we can choose free.

Then we choose such that is not prime.

Then divide this number. - May 9th 2010, 10:46 AMEinStone
So you say that we choose

and*a*arbitrarily, such that is not a prime, and then*c*needs to be chosen such that , for some*b*

Is it that what you mean? And why is this what we want?? - May 9th 2010, 11:12 AMDinkydoe
with the abc-formula we have that consists of the numbers

To make sure has the desired property we must have for some number .

That is, .

This number can ofcourse be freely chosen.

Then we must choose that satisfy this relation. Thus we must choose a such that

there exists a composite number that satisfies

wich is, the same as saying must be a composite number. Only then we can choose such that .

And we've found such that has the desired property. - May 9th 2010, 12:56 PMEinStone
Why does ac need to be composite?

- May 9th 2010, 01:01 PMDinkydoeQuote:

Why does ac need to be composite?

If prime then can not be integers. - May 9th 2010, 01:06 PMchiph588@
- May 9th 2010, 01:09 PMDinkydoe
Ah shoot, big blunder here. Shameful moment here xp

Thanks chiph

Forget everything about ac being composite. I'm sorry for this error ;p

The rest you can take for granted. - May 9th 2010, 01:11 PMEinStone
So we just need to choose a,b,c such that for some and thats all?

- May 9th 2010, 01:15 PMDinkydoe
no, I Edited my post ;p

with the abc-formula we have that http://www.mathhelpforum.com/math-he...3c97c188-1.gif consists of the numbers http://www.mathhelpforum.com/math-he...0e2c651c-1.gif

To make sure http://www.mathhelpforum.com/math-he...3c97c188-1.gif has the desired property we must have http://www.mathhelpforum.com/math-he...299727d6-1.gif for some number http://www.mathhelpforum.com/math-he...c6f135ab-1.gif.

That is, http://www.mathhelpforum.com/math-he...1ebaf4ab-1.gif.

This number http://www.mathhelpforum.com/math-he...c77a256c-1.gif can ofcourse be freely chosen.

Then we must choose http://www.mathhelpforum.com/math-he...730b61fa-1.gif that satisfy this relation. Thus we must choose a http://www.mathhelpforum.com/math-he...1dc21592-1.gif such that

there exists a number http://www.mathhelpforum.com/math-he...1cae4d4e-1.gif that satisfies http://www.mathhelpforum.com/math-he...95915071-1.gif

wich is, the same as saying http://www.mathhelpforum.com/math-he...df24373f-1.gif . Only Then we can choose http://www.mathhelpforum.com/math-he...6f662b30-1.gif such that http://www.mathhelpforum.com/math-he...82615171-1.gif.

And we've found http://www.mathhelpforum.com/math-he...dba7962e-1.gif such that http://www.mathhelpforum.com/math-he...3c97c188-1.gif has the desired property. - May 9th 2010, 01:26 PMEinStone
I don't see the difference. Which condition do you have except for .

I mean once this holds for integers a,b,c, trivially. - May 9th 2010, 01:45 PMDinkydoe
No, it's not different. It's the same, but you wanted more details right ;p?