2

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?

Riccardo.Alestra
  • 399
  • 1
    If $a$ and $c$ are large enough compared to the other coefficients then the left side will be positive for all $x,y$ so there will be no solutions. Pietro has shown that if you take the full set of equations (or even a small chunk) you get infinitely many solutions. Simple cardinality considerations show it's impossible to have an uncountable infinity of solutions, even taking all the equations. I see no question at a research level here, so I vote to close. – Gerry Myerson Feb 16 '12 at 22:24
  • 3
    Could you tell me why is this interesting? – Woett Feb 16 '12 at 22:37

1 Answers1

1

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.

Pietro Majer
  • 56,550
  • 4
  • 116
  • 260