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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: Harlan's World
Registered: 2011-05-23
Posts: 16,018

### Re: Factorising a^n + kb^n

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

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
The knowledge of some things as a function of age is a delta function.

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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: Harlan's World
Registered: 2011-05-23
Posts: 16,018

### Re: Factorising a^n + kb^n

Well,

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

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
The knowledge of some things as a function of age is a delta function.

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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: Harlan's World
Registered: 2011-05-23
Posts: 16,018

### 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
The knowledge of some things as a function of age is a delta function.

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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
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,115

### 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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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

scientia
Member
Registered: 2009-11-13
Posts: 224

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
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,115

### 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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #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
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,115

### 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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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zetafunc.
Guest

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
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,115

### 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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #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
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,115

### 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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #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
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,115

### 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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #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
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,115

### 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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #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
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,115

### 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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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