Given the Diophantine equation$$ax^2+bxy+cy^2+dx+ey+f=0$$ if the coefficients $(a,b,c,d,e,f)$ are chosen among all the prime numbers, we have infinite equations. Is it possible to prove that the solutions of the infinite equations are infinite and countable?
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closed as off topic by Gerry Myerson, Andreas Blass, Felipe Voloch, Emil Jeřábek, Pietro Majer Feb 17 '12 at 12:45Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here. If this question can be reworded to fit the rules in the help center, please edit the question. 


Yes, if e.g. $a=e=2,\, b=5,\,c=d=3,$ and $f$ varies among all primes, that equation has the solution $x=y=f$, which already makes infinitely many (distinct) solutions. 

