Apr 30, 2013 Thread starter Banned #1 P Poirot Banned Feb 15, 2012 250 Us the fact that 2 is not a quadratic residue of 3 to show that there are no integer solutions to $y^2=x^3-x+5$.
Us the fact that 2 is not a quadratic residue of 3 to show that there are no integer solutions to $y^2=x^3-x+5$.
Apr 30, 2013 #2 Bacterius Well-known member MHB Math Helper Jan 26, 2012 644 Re: show equation has no solution Take the equation modulo $3$, and you get: $$y^2 \equiv x^3 - x + 5 \equiv x^3 - x + 2 \pmod{3}$$ Now FLT tells us that $x^3 \equiv x \pmod{3}$, so we get: $$y^2 \equiv x - x + 2 \equiv 2 \pmod{3}$$ Can you finish?
Re: show equation has no solution Take the equation modulo $3$, and you get: $$y^2 \equiv x^3 - x + 5 \equiv x^3 - x + 2 \pmod{3}$$ Now FLT tells us that $x^3 \equiv x \pmod{3}$, so we get: $$y^2 \equiv x - x + 2 \equiv 2 \pmod{3}$$ Can you finish?
Apr 30, 2013 Thread starter Banned #3 P Poirot Banned Feb 15, 2012 250 Re: show equation has no solution no such y exist because 2 is a quadractic non-residue mod 3.