Diophantine Equation y cubed equals x squared plus 2

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Theorem

The indeterminate Diophantine equation:

$y^3 = x^2 + 2$

has only one solution in the Natural Numbers:

$x = 5, y = 3$


Proof

Assume that $x$ is even:

\(\ds \paren {2 k}^2 + 2\) \(=\) \(\ds 4 k^2 + 2\)
\(\ds \) \(=\) \(\ds 2 \paren {2 k^2 + 1}\)

Therefore, the right hand side is $2 \paren {2 k^2 + 1} \equiv 2 \pmod 4$

If $y$ is odd, then the left hand side will be odd:

\(\ds \paren {2 k + 1}^3\) \(=\) \(\ds 8 k^3 + 6 k^2 + 6 k + 1\)
\(\ds \) \(=\) \(\ds 2 \paren {4 k^3 + 3 k^2 + 3 k} + 1\)

and if $y$ is even, then the left hand side will be $\equiv 0 \pmod 4$

\(\ds \paren {2 k }^3\) \(=\) \(\ds 8 k^3\)
\(\ds \) \(=\) \(\ds 4 \paren {2 k^3}\)

Therefore, $x$ and $y$ must both be odd since the left hand side can never be $\equiv 2 \pmod 4$

Let us rewrite $x$ as $x = y + a$

We now demonstrate that $a$ must be even:

\(\ds x\) \(=\) \(\ds y + a\)
\(\ds \leadsto \ \ \) \(\ds 2 r + 1\) \(=\) \(\ds \paren {2 s + 1 } + 2 k\) $x$ and $y$ are both odd and $a$ is even
\(\ds \) \(=\) \(\ds \paren {2 \paren {s + k} + 1 }\)


Therefore:

\(\ds y^3\) \(=\) \(\ds \paren {y + a}^2 + 2\) substituting $x = y + a$
\(\ds \) \(=\) \(\ds y^2 + 2 a y + a^2 + 2\)
\(\ds \leadsto \ \ \) \(\ds y^3 - y^2\) \(=\) \(\ds 2 a y + a^2 + 2\) subtracting $y^2$ from both sides
\(\ds \leadsto \ \ \) \(\ds y^2 \paren {y - 1}\) \(=\) \(\ds 2 \paren {a y + \dfrac {a^2} 2 + 1}\) factoring both sides
\(\ds \) \(=\) \(\ds 2 \paren {2 k y + 2 k^2 + 1}\) $a$ is even: substituting $a = 2 k$
\(\ds \) \(=\) \(\ds 2 \paren {2 k \paren {y + k} + 1}\)

From the left hand side, we know that $y$ is odd and therefore $y - 1$ is even and the only even term on the right hand side is $2$.

Therefore, our only solution is: $y - 1 = 2 \leadsto y = 3$ and $x = 5$

$\blacksquare$


Historical Note

The Diophantine equation $y^3 = x^2 + 2$ was proved to have only the solution $x = 5, y = 3$ by Pierre de Fermat by use of the Method of Infinite Descent.

He submitted it, without proof, along with a number of others, to Pierre de Carcavi in a letter dated $14$ August $1659$.


Sources

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