Does the Equation \( y^2 = x^3 - x + 5 \) Have Integer Solutions?

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SUMMARY

The equation \( y^2 = x^3 - x + 5 \) has no integer solutions. By analyzing the equation modulo 3, it is established that \( y^2 \equiv 2 \pmod{3} \). Since 2 is a quadratic non-residue modulo 3, there are no integer values for \( y \) that satisfy the equation. This conclusion is supported by Fermat's Little Theorem, which confirms that \( x^3 \equiv x \pmod{3} \).

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Us the fact that 2 is not a quadratic residue of 3 to show that there are no integer solutions to $y^2=x^3-x+5$.
 
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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

no such y exist because 2 is a quadractic non-residue mod 3.
 

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