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?
closed as off topic by Gerry Myerson, Andreas Blass, Felipe Voloch, Emil Jeřábek, Pietro Majer Feb 17 '12 at 12:45
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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.