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 Thread  Matrix Joined: 04 Sep 2004 Posts: 1171 Location: Overflow Matrix yeah, and if you don't have a 90 degree angle in the triangle use the cosine formula: c^2 = a^2 + b^2 - 2ab * cos fi where fi is the angle of ab 17 Oct 2004, 23:19
 vbVeryBeginner Joined: 15 Aug 2004 Posts: 884 Location: \\world\asia\malaysia vbVeryBeginner eion wrote: If you want to prove that there exists three numbers a, b & c for which a^2 + b^2 = c^2 hold then one way is to just find them. For example a=3, b=4 & c=5 works, so does a=5, b=12 and c=13. em, if a^2 + b^2 = c^2 then (aX)^2 + (bX)^2 = (cX)^2 and we get infinity numbers of a b c that will conform to the a^2 + b^2 = c^2 rule Code:``` 3 4 5 3 4 5 6 8 10 9 12 15 12 16 20 27 36 45 24 32 40 ... 48 64 80 96 128 160 3 4 5 and so on... 12 16 20 5 12 13 3 4 5 10 24 26 15 20 25 20 48 52 40 96 104 and ... ``` Last edited by vbVeryBeginner on 18 Oct 2004, 00:47; edited 3 times in total 18 Oct 2004, 00:08
 Matrix Joined: 04 Sep 2004 Posts: 1171 Location: Overflow Matrix HY say, what is the purpose of this thread, i see many thing now from everywhere. it should be renamed to Math And Thoughts Thread Last edited by Matrix on 18 Oct 2004, 00:43; edited 1 time in total 18 Oct 2004, 00:21
 vbVeryBeginner Joined: 15 Aug 2004 Posts: 884 Location: \\world\asia\malaysia vbVeryBeginner sorry matrix, i know i had put many thing from everywhere (:sorry) but maybe this is the process to prove the FLT? 18 Oct 2004, 00:39
Jaques

Joined: 07 Jun 2004
Posts: 79
Location: Everywhere
Jaques
The reason why this thread was able to spread into such depth of Math is due to the fact that i explained that my formula basis can be used to find triangles specs and the confusion it generated

c = b+k
c ^ 2 = (b+k)^2
c^2 = b^2 + 2bk + k ^ 2
a^2 = 2bk + k^2
a^2 - k^2 = 2bk
(a^2 - k^2) / 2k = b
and if the difference beetween band c is a rational number k and a is a rational number then a,b,c will be rational numbers

for primes and diprimes and triprimes ect...
in the infinite set.. let p = # of primes
number of di primes = p!
number of triprimes = (p!)!
if you could figure it out for a finite set then the ramifications would be crazy you could describe primeness by knowing how many of primes and diprimes ect in the number less and the number itself then you would be able to find how many factors the number had 18 Oct 2004, 01:34  Matrix Joined: 04 Sep 2004 Posts: 1171 Location: Overflow Matrix vbVeryBeginner wrote:sorry matrix, i know i had put many thing from everywhere (:sorry) but maybe this is the process to prove the FLT? Hy, no problem, was just asking. thinking didn't hurt anybody yet 18 Oct 2004, 01:38
Eoin

Joined: 16 Jun 2003
Posts: 68
Location: Ireland
Eoin
vbVeryBeginner, thats a good point and with it you have shown that there are an infinity of numbers for which the equation holds. Again though the proof would need to be written in abstract form .

Well if anyones interested my forums at [url]board.binarynotions.com[/url]. Its all still a work in progress but if anyone has any Math, AI, fractal or other questions in the math computing area then feel free the post. I'll enjoy discussing those topics anyway. 19 Oct 2004, 14:30  vbVeryBeginner Joined: 15 Aug 2004 Posts: 884 Location: \\world\asia\malaysia vbVeryBeginner now, i came across with a big confusion. i mean, how the pithogoros knows a^2 + b^2 = c^2 ? could anyone explain to me the logic behind his algorithm? anybody know from what info he produced this algo? coz, i don't really think a^2 + b^2 is equal to c^2, i think the value of c is somehow very near to result of c^2 but not c^2 imho, the result of every "c" should be float, not something like 3^2 + 4^2 = 5^2 in round number. that means, we shouldn't be able to get round number for c. it is possible for c to contained round number? Eion wrote: Again though the proof would need to be written in abstract form why proof would need to be written in abstract form? i thought proof should be written in a very clear, easy to understand and very logical form. 23 Oct 2004, 23:14
 roticv Joined: 19 Jun 2003 Posts: 374 Location: Singapore roticv Eoin, Your emailing module is down/ not working. Hope you can get it fixed. 24 Oct 2004, 04:28
Eoin

Joined: 16 Jun 2003
Posts: 68
Location: Ireland
Eoin
vbVeryBeginner Two good questions.

To answer the second, yes of course a proof should be clear and logicial. By abstract I mean that you need to move away from using specific values and deal with abstract variables. For example your equation (aX)^2 + (bX)^2 = (cX)^2 doesn't necessarly show we can generate an infinity of numbers which fit a^2 + b^2 = c^2 until we prove that (aX)^2 + (bX)^2 = (cX)^2 will always hold. (Note <=> means "if and only if",it is used to show that one equations hold true if another one does.)
Code:
```Prove that if a^2 + b^2 = c^2 then (aX)^2 + (bX)^2 = (cX)^2.

Proof, Assume whats given ie a^2 + b^2 = c^2.

(aX)^2 + (bX)^2 = (cX)^2        <=>       (a^2 * X^2) + (b^2 * X^2) = (c^2 * X^2)         Split up powering
<=>       (a^2 + b^2) * X^2 = c^2 * X^2                   Using rule ax + bx = (a+b)x
<=>       (a^2 + b^2) = c^2                               Divide both sides by X^2
<=>       a^2 + b^2 = c^2

So now we know (aX)^2 + (bX)^2 = (cX)^2 is true if and only if a^2 + b^2 = c^2 is true.    ```
So you see above I show that if you start with a, b & c which agree with a^2 + b^2 = c^2 then mulptiplying them all by X doesn't stop them agreeing with a^2 + b^2 = c^2.

You might say well starting with a=3, b=4 & c=5 which we know work (cause we tried them) and then trying (*2) a=6, b=8 & c=10 and (*3) a=9, b=12 & c=15 that theres a pattern there and so it'll work for any (*X) a=3X, b=4X & c=5X. But in the world of proofs thats not good enough unfortunatly. Just because thee seemed to be a pattern we don't know how long it'll hold for, yeah it might work for X=6 or X=100 but we don't really know it'll work for, say, X=1,000,000 until we test it. However when you prove it as in above its proved for all X, so with the proof we do know it'll work for X=1,000,000 or X=999,999,999 or any X we can think of.

To give an example of where the pattern proof doesn't work think of primes. We know 3 is prime, we could check 5 and see yeah 5's prime, check 7 also and hey 7's prime too so there seems to be a pattern there and we could assume maybe all odd numbers are prrime. But how long will this pattern hold, and well it stops very quickly at 9 which isn't prime .

So back to question one. And remember we're talking right-angled triangles here. Yes c can be round (or an integer) but it doesn't have to be eg if a=1 and b=1 then c = sqrt(2) which isn't an integer. So what you're asking is if a=3 and b=4 on a triangle then how do we know c is exactly =5. Well the simple answer is we know it because we have a proof of it.

Think of one line, (a), of length 1 other, (b), of length 2, if we put them end to end we know that then new bigger line, (c), is of length 3. We know it cause 1+2=3. There would be a proof for that but since this is practically an "obvious truth" (We're really kinda saying (a) + (b) = (a + b) here) we don't bother with the proof. But if none of us did accept that then a proof would be needed to convince us of it. But in that light a^2 + b^2 = c^2 is not obvious, and so we don't trust it'll always hold until we see it proved which is what Pythagoras did. I posted a link earlier with a couple of versions of the proof, but if you havn't seen geometric proof before then it probably will seem quite confusing.

roticv not sure what you mean, I don't think phpbb2 has emailing built in like the vb board did. It just uses standard mailto hyperlinks i think . 24 Oct 2004, 14:05  roticv Joined: 19 Jun 2003 Posts: 374 Location: Singapore roticv Eoin, there was an error when I tried to register. Something to do with emailing to me. But somehow my account is validated. hmm vbVeryBeginner, There are many geometrical proofs to it (Eoin gave a link to it). But if you want to prove that there is integers solutions to a^2 + b^2 = c^2 it is easy too... 24 Oct 2004, 16:09
Jaques

Joined: 07 Jun 2004
Posts: 79
Location: Everywhere
Jaques
I do not mean to prove that they do exist... i find it more important to know what they are 24 Oct 2004, 17:10  vbVeryBeginner Joined: 15 Aug 2004 Posts: 884 Location: \\world\asia\malaysia vbVeryBeginner thank you again for your explanation Eion i reread the link, roticv :p (sorry) i got one more question. could i know why there are 360 degrees inside a circle? and why not 720 or infinity? 24 Oct 2004, 19:46
Jaques

Joined: 07 Jun 2004
Posts: 79
Location: Everywhere
Jaques
The explanation is a simple one there canot be infinity because the system would be useless. Babylon made this sysem and i believe it had to do with the length of the year 24 Oct 2004, 20:31  Matrix Joined: 04 Sep 2004 Posts: 1171 Location: Overflow Matrix Hello, you might be interested in this little list of Mathematicians Index of /~history/Mathematicians 01 May 2005, 12:14
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