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Topic: help i cannot figure this prob out (Read 2095 times) |
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alton pezzoli
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help i cannot figure this prob out
« on: Nov 19th, 2005, 7:10pm » |
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solve this equation and find all answers in natural numbers: X^3+3=4y(y+1)
| hidden: | [/hideb]Subtract 3 and factor to get x^3=(2y+3)(2y-1). divided the difference, 4, bec each is odd the gcd =1. So each is a perfect cube. Let
2y+3=a^3, 2y-1=b^3 so that a^3-b^3=4. this is all i understand. This problem is posted on
http://www.artofproblemsolving.com/Forum/viewtopic.php?t=61009
pls help me get the rest [hideb] |
my login is not working
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alton
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Re: help i cannot figure this prob out
« Reply #1 on: Nov 19th, 2005, 7:53pm » |
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oh also
I am having problems compiling documents in Latex
i.e. the build output will not let me click on it
Does anyone know what I am doing wrong (it’s probably something stupid)
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Eigenray
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Re: help i cannot figure this prob out
« Reply #2 on: Nov 19th, 2005, 9:10pm » |
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No two cubes of integers can differ by 4.
To go into more detail, since a3 = 2y+3 >= 5, we have a >=2. And since b<a, in fact b <= a-1, and therefore
a3-b3 >= a3-(a-1)3 = 3a(a-1)+1 >= 7 > 4.
What I don't understand is what "the build output will not let me click on it" means.
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ap
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Re: help i cannot figure this prob out
« Reply #3 on: Nov 20th, 2005, 10:27am » |
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divides the difference
D (2y+3) - (2y-1)
D 4
a^3 - b^3=4
you cant do that substitution?
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ap
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Re: help i cannot figure this prob out
« Reply #4 on: Nov 20th, 2005, 11:07am » |
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a^3-b^3 >= a^3-(a-1)^3 = 3a(a-1)+1 >= 7 > 4. That’s exactly what they got on Aops but they also have
| hidden: | [/hideb] a^3-(a-1)^3 > a^3 - b^3
b^3 > (a-1)^3
b> a-1
But for a^3-b^3 > 0, b<a then b must be between 2 consecutive integers, a contradiction. Therefore no solution [hideb] |
How did you get >=7>4 where did the 7 come from pls explain
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Eigenray
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Re: help i cannot figure this prob out
« Reply #5 on: Nov 20th, 2005, 3:05pm » |
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If a >= 2 and b<a are integers, then
a3-b3 >= 23-13 = 7 > 4,
as 1 and 8 are the closest two (positive) cubes.
on Nov 20th, 2005, 10:27am, ap wrote:you cant do that substitution?
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Eh?
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| « Last Edit: Nov 20th, 2005, 3:29pm by Eigenray » |
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ap
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Re: help i cannot figure this prob out
« Reply #6 on: Nov 20th, 2005, 5:19pm » |
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oh  i finally get it thanks
I should have looked at the solution more 
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