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Topic: system of equations (Read 1324 times) |
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Christine
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system of equations
« on: Apr 10th, 2014, 3:23pm » |
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I ran a program to find solutions to
A^3 - B^3 = x^5
A^5 - B^5 = y^3
but couldn't find any.
Perhaps somebody would like to take a shot at this.
Could anyone offer an analytical solution?
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pex
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I just wanted to share that Wolfram Alpha is spectacularly unhelpful in this case.
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Christine
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Posts: 159
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Re: system of equations
« Reply #2 on: Apr 11th, 2014, 8:05am » |
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Yeah, no kidding. I've tried Wolfram, too, at first.
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SWF
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Posts: 879
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Re: system of equations
« Reply #3 on: Apr 17th, 2014, 9:15pm » |
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That is a nice problem. I assume you are looking for the A, B, x, and y all positive integers, otherwise there are obvious trivial solutions when one or more of them are zero. There are an infinite number of non-trivial solutions too. The lowest I have found is:
A=562265004868829486
B=281132502434414743
x=43487680369
y=378999812623353224604123685177
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Christine
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Posts: 159
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Re: system of equations
« Reply #4 on: Apr 21st, 2014, 2:38pm » |
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on Apr 17th, 2014, 9:15pm, SWF wrote:| I assume you are looking for the A, B, x, and y all positive integers |
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Yes. Sorry. I forgot to mention that.
Quote:......There are an infinite number of non-trivial solutions too. The lowest I have found is:
A=562265004868829486
B=281132502434414743
x=43487680369
y=378999812623353224604123685177 |
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Thanks for providing an example.
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SWF
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Posts: 879
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Re: system of equations
« Reply #5 on: Apr 22nd, 2014, 7:49pm » |
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Here is the analytical approach I used to come up with those numbers. Start with two integers c and d, such that their difference is the 15th power of some integer:
c-d=z15
The difference of cubes or 5th powers of c and d both can be factored with c-d a factor. Define integers N and M by:
N=(c3-d3)/(c-d)
M=(c5-d5)/(c-d)
If you choose A and B as (p and q are any non-negative integers):
A=c*N15p+3M15q+10
B=d*N15p+3M15q+10
Then
A3-B3=z15N45p+10M45q+30 (a perfect 5th power)
A5-B5=z15N75p+15M75q+51 (a perfect cube)
The solution given previously is for c=2, d=1, p=0, q=0.
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Christine
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Posts: 159
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Re: system of equations
« Reply #6 on: Apr 24th, 2014, 7:58pm » |
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Awesome!
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