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 March 3rd, 2009, 09:57 AM #1 Senior Member   Joined: Nov 2007 Posts: 633 Thanks: 0 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.
 March 3rd, 2009, 10:17 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 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
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
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.

March 3rd, 2009, 12:28 PM   #4
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Re: Diophantine equation

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 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.

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!!!

 March 3rd, 2009, 03:05 PM #6 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 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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