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Find all integer solutions to ...
Find all integer solutions to
[tex]\begin{array}{l}<br /> {a^2} = a + b - 2c + 2d + e - 8\\<br /> {b^2} = - a - 2b - c + 2d + 2e - 6\\<br /> {c^2} = 3a + 2b + c + 2d + 2e - 31\\<br /> {d^2} = 2a + b + c + 2d + 2e - 2\\<br /> {e^2} = a + 2b + 3c + 2d + e - 8<br /> \end{array}[/tex]
.
I am using wolframalpha to find solutions(there are a finite number of them). I have to just find all the solution pairs that are integer.(Just type in the equations). But I have to say that all the wolfram alpha solutions are the only solutions. I want to do this by finding the number of solutions this equations have(for complex numbers). We know since that these are 5 equations with 5 unknowns, there are a finite number of solutions If I can find that this equation has let's say 32 solutions(made up number) theoretical, and I come up with 32 solutions in wolfram alpha, then I have met my goal and I can simply pick out the integer solutions.
But apparently it is too computationally complex no matter what you do. I even ran the system in MATLAB and after 3 hours, my fans on the computers started going really fast my computer shut down. So that is not going to work and I can't use brute force. But how does one try this analytically.
To all my helpers: Please don't tell me I didn't try on this problem. I have spent the whole day on it
Background:(so you know what you can help me with in the solution)
Ali Nayfeh Balakumar Balachandran Applied Nonlinear Dynamics
Strogatz Nonlinear Dynamics
Methods of mathematical Physics Courant and Hilbert
Mary Boas Mathematical Methods in Physical Sciences
Mechanics, Quantum Mechanics, Statistical Mechanics by Landau and Liftgarbagez
A Course of Pure Mathematics by G.H. Hardy
Walter Rudin Real and Complex Analysis
Walter Rudin Real analysis
Homework Statement
Find all integer solutions to
[tex]\begin{array}{l}<br /> {a^2} = a + b - 2c + 2d + e - 8\\<br /> {b^2} = - a - 2b - c + 2d + 2e - 6\\<br /> {c^2} = 3a + 2b + c + 2d + 2e - 31\\<br /> {d^2} = 2a + b + c + 2d + 2e - 2\\<br /> {e^2} = a + 2b + 3c + 2d + e - 8<br /> \end{array}[/tex]
.
The Attempt at a Solution
I am using wolframalpha to find solutions(there are a finite number of them). I have to just find all the solution pairs that are integer.(Just type in the equations). But I have to say that all the wolfram alpha solutions are the only solutions. I want to do this by finding the number of solutions this equations have(for complex numbers). We know since that these are 5 equations with 5 unknowns, there are a finite number of solutions If I can find that this equation has let's say 32 solutions(made up number) theoretical, and I come up with 32 solutions in wolfram alpha, then I have met my goal and I can simply pick out the integer solutions.
But apparently it is too computationally complex no matter what you do. I even ran the system in MATLAB and after 3 hours, my fans on the computers started going really fast my computer shut down. So that is not going to work and I can't use brute force. But how does one try this analytically.
To all my helpers: Please don't tell me I didn't try on this problem. I have spent the whole day on it
Background:(so you know what you can help me with in the solution)
Ali Nayfeh Balakumar Balachandran Applied Nonlinear Dynamics
Strogatz Nonlinear Dynamics
Methods of mathematical Physics Courant and Hilbert
Mary Boas Mathematical Methods in Physical Sciences
Mechanics, Quantum Mechanics, Statistical Mechanics by Landau and Liftgarbagez
A Course of Pure Mathematics by G.H. Hardy
Walter Rudin Real and Complex Analysis
Walter Rudin Real analysis