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Mon 14 Feb, 2005 03:55 am
well guys, this time i got a difficult riddle for you. it took me almost a week to solve it. now it is your turn, if you like to.
x^2+x = y^4 + y^3 + y^2 + y
what are all the couples of numbers (x,y) that make the equation above true. good luck!
x, y = 0, 0
x, y = -1, -1
There's probably others, but those are the most interesting!
ok! I am not thinking straight today!
If it involves x^2 and a quartic curve, then there's only two answers and I believe I found them both!
Just so you know, it only took 1 min to get on excel!!
I tried to split them up into
x (x+1) = y (y+1) (y^2 +1)
but that just shows that if any parts are 0 it's true. So it gives a couple more solutions then bizarre had, but probably still not all.
(0,0)
(0,-1)
(-1,0)
(-1,-1)
nice but the idea is to see if you find what so special about these numbers.
If I had to guess, they'd be the points on the circle with center (-.5,-.5) and radius 1/2sqrt2.
explanation, plz
or at least more details why you chose this answer/
lol, cause they looked so prettily like a circle?
I couldn't think of any other functions with points arranged like that. So just a guess...
the circle can be described by (x-.5)^2 + (y+.5)^2 = .5
That works out to x^2 + x + y^2 + y = 0
or x^2 + x = - (y^2 + y)
I'm missing a - (y^2 + 1) on the right though...
sadly, it's not the answer. for example, the point (6,2) isn't a point in the circle but if we put it in the equation, we get a true answer.
you're welcome to try again
and good job.
Are you only looking for integer solutions?
sure! sorry that i didn't mention it, i thought it was clear.
Actually, (6,2) is not a solution.
I've found:
(-6,2)
(-1,-1)
(-1,0)
(0,-1)
(0,0)
(5,2)
All solutions will be multiples of 30 (30N = 30N).
If there are more solutions, they are rather large.
then how do you explain the following thing :
Code:y^4+y^3+y^2+y=30N=30*2=60 -> there is no integer solution
"all solutions will be multiples of 30"
does not mean
"all multiples of 30 will be solutions"
Do you know of other solutions, or have you proven these are the only ones?
what i am trying to say is that ya should give some explanation why you think that's the correct answer.
As for the ordered pairs, they need no explanation. Just plug them in to the equation.
"all solutions will be multiples of 30"
Look at x^2+x mod 30 and y^4+y^3+y^2+y mod 30
The only value they have in common is 0. Therefore, for any solution, both expressions must be a multiple of 30.
I have been unable to prove that the ordered pairs listed previously are the only solutions.
Do you know of other solutions? If not, have you proven that these are the only solutions?
Well, I give up. I'm a little rusty in my algebra as it seems. I was never very good at higher grade equations anyway.
i will give you an hint :
think about following numbers multiplication. :wink:
1*2*3*4
or
4*5*6*7
and etc.
but in our case, if you look :
x^2+x=x(x+1)
do you see what i mean?
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