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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: 16,049
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: 16,049
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: 16,049
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
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
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
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
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
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
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
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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#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
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
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
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
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
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
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
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
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
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
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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