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## #1 2012-10-20 02:30:45

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 2012-10-20 05:07:32

anonimnystefy
Real Member

Offline

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

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

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón

## #3 2012-10-20 05:24:08

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 2012-10-20 07:55:01

anonimnystefy
Real Member

Offline

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

Well,

Last edited by anonimnystefy (2012-10-20 08:04:11)

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón

## #5 2012-10-20 08:00:08

zetafunc.
Guest

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

I don't understand that...

## #6 2012-10-20 08:03:57

anonimnystefy
Real Member

Offline

### 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.

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón

## #7 2012-10-20 08:12:43

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 2012-10-20 10:49:24

bobbym

Online

### 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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.

scientia
Full Member

Offline

Shouldn't it be

?

## #10 2012-10-20 22:48:08

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 2012-10-21 07:27:44

bobbym

Online

### 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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.

## #12 2012-10-21 07:38:21

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 an + kbn.

## #13 2012-10-21 07:40:22

bobbym

Online

### 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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.

zetafunc.
Guest

Oh, I see...

## #15 2012-10-21 07:45:40

zetafunc.
Guest

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

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

## #16 2012-10-21 07:47:00

bobbym

Online

### 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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.

## #17 2012-10-21 07:51:18

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 2012-10-21 07:54:55

bobbym

Online

### 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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.

## #19 2012-10-21 08:00:50

zetafunc.
Guest

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

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

## #20 2012-10-21 08:04:55

bobbym

Online

### 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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.

## #21 2012-10-21 08:17:39

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 24n+2 + 1.

## #22 2012-10-21 08:20:15

bobbym

Online

### 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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.

## #23 2012-10-21 08:24:09

zetafunc.
Guest

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

Wait, more than a century ago?!

## #24 2012-10-21 08:24:32

zetafunc.
Guest

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

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

## #25 2012-10-21 08:26:51

bobbym

Online

### 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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.