ATAR Notes: Forum

VCE Stuff => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematics => Topic started by: Martoman on April 08, 2010, 03:32:54 am

Title: proof by contradiction? or other method?
Post by: Martoman on April 08, 2010, 03:32:54 am: Show that $n^4 -20n^2 + 4$ is composite for all n when n is an integer.

Just a verification required

Assume $n^4 -20n^2 + 4$ is prime. Then some prime p, $p| n^4 -20n^2 + 4$

$\implies \frac{n^4 -20n^2 + 4}{p}$ which means $\frac{n^4-20n^2}{p} + \frac{4}{p}$

This however means that $p = +-2$ as the only prime number to divide 4 in such a way that it produces an integer answer... because you can't have a composite or prime decimal as far as I know. Having p restricted in such a way, means that the $\frac{n^4 -20n^2 + 4}{p}$ will be a fraction for some n (try n = 1) then $\frac{-15}{\pm2}$ isn't a natural number. Hence, a contradiction is reached assuming that it was prime to start with.
Hence, it must be composite.

*edit* 1...is..not...a...prime....number....

*edit twice, mainly because you dont_need_to_factorise... i factorized for whatever reason (ie: it was cool to do so)*
Title: Re: proof by contradiction? or other method?
Post by: brightsky on April 08, 2010, 12:32:26 pm: Wouldn't it be easier if you just factorised it?

$n^4 - 20n^2 + 4 = n^4 - 4n^2 + 4 - 16n^2 = (n^2 - 2)^2 - (4n)^2 = (n^2 -4n - 2)(n^2 + 4n - 2)$ .

Hence, it is a composite number.
Title: Re: proof by contradiction? or other method?
Post by: the.watchman on April 08, 2010, 12:49:28 pm: It may be worth noting that when solving the simultaneous equations:

$n^2-4n+2 = n^4 - 20n^2+4$ AND $n^2+4n-2=1$

This does not yield integer solutions (nor does the opposite), thus suggesting that the factors cannot ONLY be 1 or itself when n is an integer

But proof by contradiction is way cooler 8-)
Title: Re: proof by contradiction? or other method?
Post by: brightsky on April 08, 2010, 12:59:42 pm: Why can't it be?

$(n^2 - 4n - 2)(n^2 + 4n - 2) = 1 (n^2 - 4n - 2)(n^2 + 4n - 2) = 1 \cdot (n^4 - 20n^2 + 4)$ .
Title: Re: proof by contradiction? or other method?
Post by: Martoman on April 08, 2010, 01:00:07 pm: Quote from: brightsky on April 08, 2010, 12:32:26 pm
Wouldn't it be easier if you just factorised it?

$n^4 - 20n^2 + 4 = n^4 - 4n^2 + 4 - 16n^2 = (n^2 - 2)^2 - (4n)^2 = (n^2 -4n - 2)(n^2 + 4n - 2)$ .

Hence, it is a composite number.

mmm see this is me not thinking what a prime number is. My argument works, although I like brightsky's as well.
Title: Re: proof by contradiction? or other method?
Post by: the.watchman on April 08, 2010, 01:01:17 pm: Quote from: brightsky on April 08, 2010, 12:59:42 pm
Why can't it be?

$(n^2 - 4n - 2)(n^2 + 4n - 2) = 1 (n^2 - 4n - 2)(n^2 + 4n - 2) = 1 \cdot (n^4 - 20n^2 + 4)$ .

I meant cannot ONLY be, sorry for confusion :P
Title: Re: proof by contradiction? or other method?
Post by: brightsky on April 08, 2010, 01:04:00 pm: Quote from: the.watchman on April 08, 2010, 01:01:17 pm
Quote from: brightsky on April 08, 2010, 12:59:42 pm
Why can't it be?

$(n^2 - 4n - 2)(n^2 + 4n - 2) = 1 (n^2 - 4n - 2)(n^2 + 4n - 2) = 1 \cdot (n^4 - 20n^2 + 4)$ .

I meant cannot ONLY be, sorry for confusion :P

Ahh, gotcha. ;D
Title: Re: proof by contradiction? or other method?
Post by: the.watchman on April 08, 2010, 01:10:54 pm: Quote from: AzureBlue on April 08, 2010, 01:05:53 pm
Quote from: the.watchman on April 08, 2010, 12:49:28 pm
But proof by contradiction is way cooler 8-)
+1

Lol :)
Anyone got any good resources where I can get questions to practice? Or induction questions :)
Title: Re: proof by contradiction? or other method?
Post by: Martoman on April 08, 2010, 01:15:17 pm: Yes the art and craft of problem solving. I'm working through this now thanks to true tears. It has some really nice questions in it.

and err, is my proof right??? No one has answered this?

*edit* not to double post
Title: Re: proof by contradiction? or other method?
Post by: the.watchman on April 08, 2010, 01:19:01 pm: Quote from: Martoman on April 08, 2010, 01:15:17 pm
Yes the art and craft of problem solving. I'm working through this now thanks to true tears. It has some really nice questions in it.

Good, I shall find this and try it. :)
Title: Re: proof by contradiction? or other method?
Post by: Martoman on April 08, 2010, 01:20:36 pm: TST = team selection tests for IMO?
..... epic.
Title: Re: proof by contradiction? or other method?
Post by: GerrySly on April 08, 2010, 01:27:07 pm: Quote from: Martoman on April 08, 2010, 01:20:36 pm
TST = team selection tests for IMO?
..... epic.

You working to the IMO or just fulfilling a craving for maths?
Title: Re: proof by contradiction? or other method?
Post by: GerrySly on April 08, 2010, 01:45:21 pm: The question was actually directed at Martoman, I know you are going for the IMO most of your posts are about getting there lol
Title: Re: proof by contradiction? or other method?
Post by: humph on April 08, 2010, 02:13:21 pm: Quote from: the.watchman on April 08, 2010, 12:49:28 pm
But proof by contradiction is way cooler 8-)
DISAGREE

Half the time it's not necessary anyway (as in this case) and can just lead to confusion.
See Tim Gowers' thoughts:
http://gowers.wordpress.com/2010/03/28/when-is-proof-by-contradiction-necessary/
Title: Re: proof by contradiction? or other method?
Post by: the.watchman on April 08, 2010, 02:16:46 pm: Ok, I get your point, but I've just learnt them recently :P
Title: Re: proof by contradiction? or other method?
Post by: Martoman on April 08, 2010, 02:51:49 pm: Quote from: GerrySly on April 08, 2010, 01:45:21 pm
The question was actually directed at Martoman, I know you are going for the IMO most of your posts are about getting there lol

mmmm, i've been asked to go further than AMC, but that requires effort.
Title: Re: proof by contradiction? or other method?
Post by: Martoman on April 08, 2010, 02:56:37 pm: Quote from: AzureBlue on April 08, 2010, 02:53:21 pm
Quote from: Martoman on April 08, 2010, 02:51:49 pm
Quote from: GerrySly on April 08, 2010, 01:45:21 pm
The question was actually directed at Martoman, I know you are going for the IMO most of your posts are about getting there lol
mmmm, i've been asked to go further than AMC, but that requires effort.
How much further? lol
AMC requires effort too ;)

AMC requires effort?????? well I give it credit in some beautiful questions that have been produced by it :)

Further meaning whatever comes after it?? I don't know what this is. I assume more testing. Then more. Then more.
Title: Re: proof by contradiction? or other method?
Post by: Martoman on April 08, 2010, 03:03:33 pm: I've only been able to participate in it a few times due to my knack to not turn up to school for a few weeks at random intervals of the year. In those times i have done it, I'm sorry, but haven't. From the most recent past papers however, with the knowledge I now have, *some* of the questions are laughable.
Title: Re: proof by contradiction? or other method?
Post by: Martoman on April 08, 2010, 03:08:51 pm: mmmm the algebra ones are... but... I STILL can't get q30 from 2008.
Title: Re: proof by contradiction? or other method?
Post by: Martoman on April 08, 2010, 03:10:51 pm: Senior, the one with >9000 square roots.
Title: Re: proof by contradiction? or other method?
Post by: brightsky on April 08, 2010, 04:51:45 pm: What was the question? I'm interested. ;)
Title: Re: proof by contradiction? or other method?
Post by: the.watchman on April 08, 2010, 04:54:47 pm: Q30 2008 AMC, trying it now sir! :P
Title: Re: proof by contradiction? or other method?
Post by: Martoman on April 08, 2010, 05:09:34 pm: LOL chinese.
Title: Re: proof by contradiction? or other method?
Post by: TrueTears on April 08, 2010, 06:13:34 pm: Quote from: the.watchman on April 08, 2010, 01:19:01 pm
Quote from: Martoman on April 08, 2010, 01:15:17 pm
Yes the art and craft of problem solving. I'm working through this now thanks to true tears. It has some really nice questions in it.

Good, I shall find this and try it. :)
I sent it to u last time :S

and yes art and craft fucking rox, cept for the number theory part, you should read a much more elementary text b4 tackling the ones in anc
Title: Re: proof by contradiction? or other method?
Post by: the.watchman on April 08, 2010, 08:08:06 pm: Quote from: TrueTears on April 08, 2010, 06:13:34 pm
Quote from: the.watchman on April 08, 2010, 01:19:01 pm
Quote from: Martoman on April 08, 2010, 01:15:17 pm
Yes the art and craft of problem solving. I'm working through this now thanks to true tears. It has some really nice questions in it.

Good, I shall find this and try it. :)
I sent it to u last time :S

and yes art and craft fucking rox, cept for the number theory part, you should read a much more elementary text b4 tackling the ones in anc

oh, that's right
i've been so busy i haven't had time to read them yet :P
Title: Re: proof by contradiction? or other method?
Post by: TrueTears on April 08, 2010, 11:33:09 pm: Quote from: AzureBlue on April 08, 2010, 05:05:51 pm
If the question I posted previously was the right one, here's the solution (but in Chinese unfortunately).
ta ma de... zhen li hai.... wo cao...
Title: Re: proof by contradiction? or other method?
Post by: brightsky on April 08, 2010, 11:35:40 pm: Quote from: TrueTears on April 08, 2010, 11:33:09 pm
Quote from: AzureBlue on April 08, 2010, 05:05:51 pm
If the question I posted previously was the right one, here's the solution (but in Chinese unfortunately).
ta ma de... zhen li hai.... wo cao...
hahaha, di que hen li hai. zai AMC jing sai zhong, na'er hui xiang chu na ma jing cai de da an?
Title: Re: proof by contradiction? or other method?
Post by: TrueTears on April 08, 2010, 11:38:16 pm: xian dai shu xue zheng de shi zhong guo ren de tian xia...
Title: Re: proof by contradiction? or other method?
Post by: brightsky on April 08, 2010, 11:40:46 pm: I agree.
Title: Re: proof by contradiction? or other method?
Post by: Gloamglozer on April 09, 2010, 12:01:08 am: Ah, Pin Yin. If only VN was like Word where you type the Pin Yin and the character appears. :P