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#1 2012-10-19 03:30:45

zetafunc.
Guest

Factorising a^n + kb^n

Is there a method of deducing whether or not a polynomial of the form

is factorisable?

For instance, we can see that

and

but something like

is not factorisable.

Is there a way to tell if we can factorise something of this form? Is there an easy way to do this, or would your best bet be just to write a general factorisation and solve for your general co-efficients? (for instance, writing one factor as (x^2 + ax + b) and finding a and b or something.

#2 2012-10-19 06:07:32

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

Re: Factorising a^n + kb^n

You can try cyclotomic polynomials. I remember bobbym said they can be used for deriving such identities, so...


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#3 2012-10-19 06:24:08

zetafunc.
Guest

Re: Factorising a^n + kb^n

Yes, I saw that too, but I have no idea how to use those here. I can't find anything that discusses multivariable cyclotomic polynomials.

#4 2012-10-19 08:55:01

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

Re: Factorising a^n + kb^n

Well,

Last edited by anonimnystefy (2012-10-19 09:04:11)


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“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#5 2012-10-19 09:00:08

zetafunc.
Guest

Re: Factorising a^n + kb^n

I don't understand that...

#6 2012-10-19 09:03:57

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,507

Re: Factorising a^n + kb^n

Fixed it. You can treat a/b like only one variable, and then calculate ((a/b)+k)^n and then multiply by b^n.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#7 2012-10-19 09:12:43

zetafunc.
Guest

Re: Factorising a^n + kb^n

Are you sure that is correct? I tested it with Sophie-Germain's identity (k = 4, n = 4) and I am not getting a^4 + 4b^4... unless I went wrong somewhere.

#8 2012-10-19 11:49:24

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,261

Re: Factorising a^n + kb^n

Hi all;

I am not getting that either.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#9 2012-10-19 22:22:49

scientia
Member
Registered: 2009-11-13
Posts: 222

Re: Factorising a^n + kb^n

Shouldn't it be

?

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#10 2012-10-19 23:48:08

zetafunc.
Guest

Re: Factorising a^n + kb^n

That is correct -- but I'm more interested in integer factors. In other words, the above is not desirable as a does not always divide b (for a = 2, b = 3 for instance). For example, suppose that I wanted to deduce whether or not something was prime -- finding a factorisation with fractions in it might not help.

#11 2012-10-20 08:27:44

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,261

Re: Factorising a^n + kb^n

Hi;

Perhaps the above form is suggesting that a factorization only occurs when b divides a?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#12 2012-10-20 08:38:21

zetafunc.
Guest

Re: Factorising a^n + kb^n

Why, though? I can't see why, for example, if a² - b² factorises to (a-b)(a+b), that one condition is that b divides a. Yet that is also of the form a[sup]n[/sup] + kb[sup]n[/sup].

#13 2012-10-20 08:40:22

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,261

Re: Factorising a^n + kb^n

Hi;

That was a little bit of mathematical humor.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#14 2012-10-20 08:44:55

zetafunc.
Guest

Re: Factorising a^n + kb^n

Oh, I see...

#15 2012-10-20 08:45:40

zetafunc.
Guest

Re: Factorising a^n + kb^n

Do you still have that computer program you used to compute factorisations?

#16 2012-10-20 08:47:00

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,261

Re: Factorising a^n + kb^n

Hi;

I threw it away in favor of a better program!


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#17 2012-10-20 08:51:18

zetafunc.
Guest

Re: Factorising a^n + kb^n

So you were able to pull that a^6 + 8b^6 factorisation off the top of your head?

#18 2012-10-20 08:54:55

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,261

Re: Factorising a^n + kb^n

Hi;

Of course not:

To start, did you read post #2?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#19 2012-10-20 09:00:50

zetafunc.
Guest

Re: Factorising a^n + kb^n

Yes but you never told me how you used cyclotomic polynomials to do that...

#20 2012-10-20 09:04:55

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,261

Re: Factorising a^n + kb^n

More than a century ago, I came across a book with a big chart of Aurifeuillian Factorizations. I was amazed.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#21 2012-10-20 09:17:39

zetafunc.
Guest

Re: Factorising a^n + kb^n

How does that relate to this? The wiki article is saying it is a factorisation of the form 2[sup]4n+2[/sup] + 1.

#22 2012-10-20 09:20:15

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,261

Re: Factorising a^n + kb^n

The point is I had tables of them like a table of integrals or sums.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#23 2012-10-20 09:24:09

zetafunc.
Guest

Re: Factorising a^n + kb^n

Wait, more than a century ago?!

#24 2012-10-20 09:24:32

zetafunc.
Guest

Re: Factorising a^n + kb^n

So, I am guessing this book might be a bit difficult for me to find...

#25 2012-10-20 09:26:51

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 85,261

Re: Factorising a^n + kb^n

Yes, I am ancient. You know that old quote:

The first hundred years is the hard part, after that it is all clear sailing.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

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