I yield my turn. I am a bit busy right now. Go ahead Galathaea
Printable View
I yield my turn. I am a bit busy right now. Go ahead Galathaea
Well, I was working on this one problem the other day, and it actually surprised me when I figured out the answer, so maybe some of you would like to check it out.
Let n be a given positive integer. How many solutions are there in ordered positive integer pairs (x, y) to the equation
In other words, finds some equation based on n and / or properties of n that calculates the number of solutions to this equation. Both x and y are positive numbers, and the fact that the pair is ordered means that when x and y are different (x, y) is counted as a separate solution from (y, x).Code:xy
----- = n
x + y
Number of pairs = (number of divisors of n) * 2 - 1Quote:
Originally posted by galathaea
finds some equation based on n and / or properties of n that calculates the number of solutions to this equation.
e.g. if n=12, divisors are 1, 2, 3, 4, 6, 12. There are 6 of them. So there are 11 ordered pairs (x,y) solving xy/(x+y)=12.
There are more than 11 solutions to the n=12 subproblem, so your solution is off. I would give the total for n=12, maybe even enumerate them, but I'll give some more time first for you track down your error (because I already think this will be solved soon from your post :)).
Oh that's right... there was one part where I had if this then do that, and if the other thing, I'll come back to that later but i forgot ;).
Anyway, I already threw out the paper I did it on, so maybe I'll go through my steps again later.
Hmmm, k let's try this:
first, prime factor n:
n=a^i*b^j*c^k*d^l*... and so on.
Now it should be:
(let NS[n] denote "the number of paired solutions for n")
NS[n]=NS[a^i]*NS[b^j]*NS[c^k]*NS[d^l]*...
=(2i+1)*(2j+1)*(2k+1)*(2l+1)*...
This means there are 15 (not 11!) solutions for n=12, for example:
12 = 2^2*3^1
NS[12]=NS[2^2]*NS[3^1]
=(2*2+1)*(2*1+1)=(5)(3)=15.
Here's a new question:
good luck!Code:a+b+c=-6
ab+ac+bc=2
a^3+b^3+c^3=3
a=?, b=?, c=?, abc=?
If I'm correct, that is not soluble over integers (a check over the possibilities mod 7), so are you looking for solution over a radical field?
And yes, your previous solution was correct even though I don't quite see your reasoning... :D
That's right, there are no integer solutions.Quote:
Originally posted by galathaea
If I'm correct, that is not soluble over integers (a check over the possibilities mod 7), so are you looking for solution over a radical field?
My mind works in ways even I can't comprehend.Quote:
Originally posted by galathaea
And yes, your previous solution was correct even though I don't quite see your reasoning...
Well, a quick expansion of (a+b+c)^3 leads me to abc = 61. This then gives me the fact that a, b, and c can be found as the roots to x^3 + 6 x^2 + 2 x - 61 = 0 (using either the cubic equation root formula or elliptic function methods can give the precise form of the radical -- or something like Mathematica -- but from here it is just rote). Unless I made an error in my multinomial expansion...
I'm sorry, I meant to ask for just abc=?. You got it correct, with 61. It's clear from the problem set-up that a, b, and c are symmetrical or interchangeable (what's the correct term for this?), so obviously there is no single correct answer, but perhaps it would still be possible to find numbers that fit the bill. Although, trying to find three numbers whose product is prime may be quite annoying.Quote:
Originally posted by galathaea
Well, a quick expansion of (a+b+c)^3 leads me to abc = 61. This then gives me the fact that a, b, and c can be found as the roots to x^3 + 6 x^2 + 2 x - 61 = 0 (using either the cubic equation root formula or elliptic function methods can give the precise form of the radical -- or something like Mathematica -- but from here it is just rote). Unless I made an error in my multinomial expansion...
In a completely different and strange universe, there exists an earth divided into three human races or tribes. These tribes have been forever in epic battle, and are fearful and hating of each other. One group is known as the Hashishim and are ruled by an enigmatic man known as Hassan-i-Sabbah. They claim that nothing is true, and that everything is therefore permissible. Another group is known as the Bilderbergers and is under the control of a man named John Birch who vehemently denies the views of the Hashishim and instead declares that everything is true, yet nothing permissible. The final (and least populous) group calls themselves the Discordians and, although technically not ruled, pay direct homage to their spiritual leader Korzybski. The Discordians make no claims and just generally feel that reality is really confusing.
Long ago, 3 obtuse-triangular fragments of an ancient and enigmatic metal alloy engraving were found, with strange words in a strange language enscribed upon them. These 3 fragments fell one each into the hands of the tribes. Each tribe soon developed its own mythology as to the origins of the great engraving. The Hashishim felt it was a part of the Necronomicon, composed from the elemental universe by Abdul Alhazrad on one of his trips to Sirius. The Bilderbergers felt it was a Merovingian engraving detailing plans to arrive to Sirius. And the Discordians felt it was a direct gift from the peoples of Sirius to the Dogon, who passed it on to the Discordians for their continuing support. All myths led to the conclusion that the race of people that finds "the number of true and distinct meaning words" and utters this number at a huge obelisk at the north pole will be granted eternal life and happiness, and they all agree on this because the Great Obelisk of the North has
this written on it in a common tongue of the peoples. It also mentioned "horrible destructions" for error, and so no person was allowed near in fear of error.
There was a famous philosopher of ancient times, called Hermes Trismegistus and of unknown tribal origin, who laid out the basics of the analysis of the fragments. He first pointed out that there are 5 syllables from which all the words are constructed: mo, lo, me, le, and ni. Using his great philosophy, he determined that the syllable ni was meaning-neutral. By this, he meant that the words lomo and lomoni have exacly the same meaning, as do nilomo, loninimo, and so on. Therefore, although there were thousand of words enscribed upon the fragments, not all of the words were distinct. It was realized that this was the real task at hand.
Soon after the philosophies of Hermes were published, the three tribes became suspicious of each other and no longer allowed public analysis of the fragments. All copies and notes were burned in a worldwide conflagration of paranoia. A popular front of like-minded peoples from all three tribes attempted to "return the knowledge to the masses" and was able through their secret geometries to discover some other meaning-neutral sequences of syllables, which eventually became popular knowledge. They discovered that "mome", "memo", "lole", and "lelo" all were meaning neutral when found in words. Eventually, though, all members of the popular front were either killed or called very bad names, and the front disbanded and became only history.
Each of the three tribes pursued their analyses in private. The Hashishim perfected cryptanalysis and applied it to their fragment. They discovered that "momo" has the same meaning as "me". The Bilderbergers used the ancient analysis of kabbalah to discern that "lo" has the same meaning as "le". And finally, through application of their advanced methods in pattern recognition, the Discordians were able to find out that "molomolomolo" is meaning-neutral. Among the three of them they had enough information to speak unto the obelisk the correct number of distinct words in the fragments, but they dared not share their information for fear the others would learn the answer before themselves.
Then, after many years of secrecy, a magical little girl Guzel was born, the daughter of three parents (one from each of the tribes). Seeing that she was of all races, and thus could benefit all equally, the three tribes gathered at a great convention known as the Trilateral Commission, and gave the little girl the secrets they had learned. Then Guzel and her pet Xeon went forth on a great journey to the Great Obelisk of the North during which she had to use the knowledge to determine what to speak. What number of distinct words should she determine there are in the fragments?
mmm...Quote:
What number of distinct words should she determine there are in the fragments?
Can be that she could not determine nothing because she was eaten by her pet? :D ;)
3!Quote:
Originally posted by galathaea
It also mentioned "horrible destructions" for error, and so no person was allowed near in fear of error.
...
What number of distinct words should she determine there are in the fragments?
Hey, you said there would be horrible destruction... where is the horrible destruction!