March 3rd, 2009, 10:57 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, 11:17 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, 01:24 PM  #3  
Senior Member Joined: Nov 2007 Posts: 633 Thanks: 0  Re: Diophantine equation Quote:
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, 01:28 PM  #4  
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 Quote:
 
March 3rd, 2009, 01:50 PM  #5  
Senior Member Joined: Nov 2007 Posts: 633 Thanks: 0  Re: Diophantine equation Quote:
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, 04:05 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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