Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ ¹ ² ³ °
 

You are not logged in. #1 20121203 12:53:37
A few questionsWell, I will be posting a lot of questions in this thread one by one, after completely clearing my doubts in each of them. 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #2 20121203 14:25:34
Re: A few questions
Last edited by scientia (20121203 14:31:30) #3 20121203 14:38:39
Re: A few questionsThank You  That was a nice one Prove that (a+b) is a perfect square 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #4 20121203 21:05:09
Re: A few questionsHi Agnishom; are of the form: that is proved by expanding using algebra. Substituting integers into k, m and n will give every solution. So If you make k equal to one then Since they have no factors in common the GCD ( m (m+n), n (m+n), mn) = 1 and so GCD(a,b,c) is 1. So a + b is a square for every solution. We are done. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #5 20121203 23:09:29
Re: A few questionsI understood but how about the proof of that? 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #6 20121203 23:19:32
Re: A few questionsHi Agnishom; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #7 20121203 23:28:07
Re: A few questionsNo I like your proof. But how do I prove this: ??? 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #8 20121203 23:42:09
Re: A few questionsHi; Multiply through by k: Combine the LHS. After division by ( m + n ) in the numerator and denominator. So In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #10 20121204 00:11:14
Re: A few questionsBobbym is only trying to link to an image but the server on which the image is hosted isn't allowing remote linking without your first visiting the site. #11 20121204 00:14:28
Re: A few questionsHi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #12 20121204 00:22:05
Re: A few questionsAgnishom is seeing this image:
You're welcome. Last edited by scientia (20121204 00:23:03) #13 20121204 00:28:24
Re: A few questionsHi Scientia; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #14 20121204 00:28:56
Re: A few questionsYou got me wrong... 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #16 20121204 00:37:35
Re: A few questionsHi Bob;
Not yet. I honestly was at a bonafide site that I googled for that gif. It seems no matter how smart I think I am and how long I have lived there is still a billion tricks I have never even heard of. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #17 20121204 00:40:26
Re: A few questionsHow can you say that every solution can be expressed in that form? 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #18 20121204 00:46:04
Re: A few questionsHi Agnishom; then we can say a = k m (m+n) and b = k n (m+n) and c = k m n where k,m, and n are arbitrary integers that do not equal 0. Also, ( m + n ) does not equal 0. Of course this does not apply here because everything is positive. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #19 20121204 00:51:32
Re: A few questionsBut how do you get that form??? 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #20 20121204 00:54:09
Re: A few questionsHi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #21 20121204 12:37:57
Re: A few questionsI agree with bobbym that you don't need to reprove a result every time you want to use it, but if someone wants to use a result that you have never seen before and doesn't provide a proof of it then it is reasonable to be unhappy with the result, as Agnishom is. Let l = gcd(a,b), m = gcd(a,c), n = gcd(b,c) Since gcd(a,b,c) = 1, it follows that gcd(l,m) = gcd(l,n) = gcd(m,n) = 1 and so there exist positive integers a', b', c' such that a = lma' b = lnb' c = mnc' Also gcd(a', b') = gcd(a', c') = gcd(b', c') = gcd(a', n) = gcd(b', m) = gcd(c', l) = 1. Now if we substitute our expressions for a, b, c into the original equation we obtain Since gcd(c', la'b') = 1, it follows that c' = 1, and hence ma' +nb' = la'b'. Also gcd(a', nb') = gcd(b', ma') = 1, so a' = b' = 1 and so l = m + n. Therefore a = m(m+n), b = n(m+n), c = mn. Of course we also have that gcd(m,n) = 1, but we don't need to use this to see that a + b = (m + n)^2. #22 20121204 13:44:51
Re: A few questionsHi;
I did say I would provide the derivation of this if he still wanted to see it. What I am saying is it is an identity for positive integers. This is proved through the algebra in post #8. Where it comes from is not really important. If I had guessed at it and then proved it that would be fine. I could use it as a theorem in Agnishom's question. Perhaps post#8 is not rigorous enough, but if it is then it holds for all k,m and n that are positive integers because it is an identity. ( Actually it holds for a great deal more but we only need the positive integers. ) To generate all the solutions just means having to iterate k, m and n for 1,2,3,4,5.... sort of like a loop inside a loop inside a loop. 3 nested loops, a structure Agnishom is familiar with I am sure. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #23 20121204 14:31:41
Re: A few questionsThank You, Avon
Yes I wanted to see it, and Avon's post has helped. 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #24 20121204 15:10:41
Re: A few questionsHi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #25 20121204 15:56:35
Re: A few questionsbobbym, I apologise if I have caused any offence but I think you are missing something. 