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March 3rd, 2009, 09:57 AM   #1
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Diophantine equation

x,y,z,a,b,c integers >1

xy=az + b (1)

x+y+z=c (2)

a,b,c are known

Can we solve the equation?

Thank you for any help.
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March 3rd, 2009, 10:17 AM   #2
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Re: Diophantine equation

Yes, it can be solved.

xy + ax + ay = ac + b

It depends on the factorization of a^2 + ac + b:
http://www.alpertron.com.ar/METHODS.HTM#SHyperb
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March 3rd, 2009, 12:24 PM   #3
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Re: Diophantine equation

Quote:
Originally Posted by CRGreathouse
Yes, it can be solved.

xy + ax + ay = ac + b

It depends on the factorization of a^2 + ac + b:
http://www.alpertron.com.ar/METHODS.HTM#SHyperb
Thank you for your reference.
I read it quickly but it seems more complicated case if a, b, and c are big big numbers (300 or 400 digits).
We will have a lot of computing work to do before finding any solutions.
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March 3rd, 2009, 12:28 PM   #4
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Re: Diophantine equation

Quote:
Originally Posted by momo
Thank you for your reference.
I read it quickly but it seems more complicated case if a, b, and c are big big numbers (300 or 400 digits).
We will have a lot of computing work to do before finding any solutions.
If a, b, and c are positive and at least 300 digits, you'll have trouble solving this since you need to factor a^2 + ac + b. (You can't get around this: solutions to the problem would give away the factorization.) Even if a is just 200 digits you have a C400 which is hard.
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March 3rd, 2009, 12:50 PM   #5
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Re: Diophantine equation

Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by momo
Thank you for your reference.
I read it quickly but it seems more complicated case if a, b, and c are big big numbers (300 or 400 digits).
We will have a lot of computing work to do before finding any solutions.
If a, b, and c are positive and at least 300 digits, you'll have trouble solving this since you need to factor a^2 + ac + b. (You can't get around this: solutions to the problem would give away the factorization.) Even if a is just 200 digits you have a C400 which is hard.
I knew that it will be too hard with big numbers.
a,b and c are positive.
We do not know for sure if a^2 + ac + b is easily factorizable.
Even if .. how to compute all the factors?
If a^2 + ac + b is prime?

So thanks a lot!!!
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March 3rd, 2009, 03:05 PM   #6
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Re: Diophantine equation

When a^2 + ac + b is prime, it's easy -- there are fast checks for numbers with even a thousand digits. But when it has several large factors, it's extremely hard to work with.
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