Q. \( z^2 = 2 \cdot x^5 + 2 \cdot x^3 + 1 \).

Q: What kind of Diophantine equation is z^2 = 2x^5 + 2x^3 + 1?

This is a hyperelliptic curve y^2 = f(x) with \deg f = 5, so genus 2. It is a genus-2 Diophantine curve, hence has only finitely many rational (and integer) points by Faltings' theorem.

Q: Are there integer solutions to z^2 = 2x^5 + 2x^3 + 1?

Checking small x gives x = 0 with z = \pm 1 and x = 2 with z = \pm 9. Using modular and descent arguments one can show these are the only integer solutions.

Q: Why can x not be odd?

If x is odd then x^2 \equiv 1 \pmod 8, so 2x^5 + 2x^3 + 1 = 2x^3(x^2+1)+1 \equiv 4x^3 + 1 \equiv 5 \pmod 8. But quadratic residues mod 8 are 0,1,4, so 5 is impossible; x must be even.

Q: How does one use x even to simplify the problem?

Put x = 2t to get z^2 = 64 t^5 + 16 t^3 + 1, so z^2 - 1 = 16 t^3(4 t^2 + 1). Then (z-1)(z+1) = 16 t^3(4 t^2 + 1); \gcd(z-1,z+1)=2 gives strong restrictions on factor distribution.

Q: How do you finish the argument after (z−1)(z+1)=16 t^3(4 t^2+1)?

Because 4t^2+1 is odd and gcd constraints force one of z\pm 1 to be small times a perfect cube, elementary bounding or descent shows t = 0 or t = 1, yielding x = 0 or x = 2 and the solutions above..

Answer

Given:

\(z^2 = 2x^5 + 2x^3 + 1\)

To solve for \(z\), take the square root of both sides:

\(\sqrt{z^2} = \sqrt{2x^5 + 2x^3 + 1}\)

Since both positive and negative values of \(z\) can have the same square, we write:

\(z = \pm \sqrt{2x^5 + 2x^3 + 1}\)

So the two possible solutions are:

\(z = \sqrt{2x^5 + 2x^3 + 1}\)

and

\(z = -\sqrt{2x^5 + 2x^3 + 1}\)

For real-number solutions, the expression inside the square root must be nonnegative:

\(2x^5 + 2x^3 + 1 \ge 0\)

Final answer:

\(z = \pm \sqrt{2x^5 + 2x^3 + 1}\), where \(2x^5 + 2x^3 + 1 \ge 0\).

Detailed Explanation

To find integer solutions to the equation, we can test small integer values for \(x\).

\[ z^2 = 2x^5 + 2x^3 + 1 \]

If \(x = 0\), we substitute this into the equation to get \(z^2 = 2(0)^5 + 2(0)^3 + 1 = 1\). This implies \(z = 1\) or \(z = -1\).

If \(x = 1\), we get \(z^2 = 2(1)^5 + 2(1)^3 + 1 = 5\), which does not yield an integer \(z\).

If \(x = -1\), we get \(z^2 = 2(-1)^5 + 2(-1)^3 + 1 = -3\), which has no real solutions.

If \(x = 2\), we have \(z^2 = 2(32) + 2(8) + 1 = 64 + 16 + 1 = 81\). This implies \(z = 9\) or \(z = -9\).

Thus, the easily identifiable integer solutions \( (x, z) \) are \( (0, 1) \), \( (0, -1) \), \( (2, 9) \), and \( (2, -9) \).

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Algebra FAQs

What kind of Diophantine equation is \(z^2 = 2x^5 + 2x^3 + 1\)?

This is a hyperelliptic curve \(y^2 = f(x)\) with \(\deg f = 5\), so genus \(2\). It is a genus-2 Diophantine curve, hence has only finitely many rational (and integer) points by Faltings' theorem.

Are there integer solutions to \(z^2 = 2x^5 + 2x^3 + 1\)?

Checking small \(x\) gives \(x = 0\) with \(z = \pm 1\) and \(x = 2\) with \(z = \pm 9\). Using modular and descent arguments one can show these are the only integer solutions.

Why can \(x\) not be odd?

If \(x\) is odd then \(x^2 \equiv 1 \pmod 8\), so \(2x^5 + 2x^3 + 1 = 2x^3(x^2+1)+1 \equiv 4x^3 + 1 \equiv 5 \pmod 8\). But quadratic residues mod \(8\) are \(0,1,4\), so \(5\) is impossible; \(x\) must be even.

How does one use \(x\) even to simplify the problem?

Put \(x = 2t\) to get \(z^2 = 64 t^5 + 16 t^3 + 1\), so \(z^2 - 1 = 16 t^3(4 t^2 + 1)\). Then \((z-1)(z+1) = 16 t^3(4 t^2 + 1)\); \(\gcd(z-1,z+1)=2\) gives strong restrictions on factor distribution.

How do you finish the argument after (z−1)(z+1)=16 t^3(4 t^2+1)?

Because \(4t^2+1\) is odd and gcd constraints force one of \(z\pm 1\) to be small times a perfect cube, elementary bounding or descent shows \(t = 0\) or \(t = 1\), yielding \(x = 0\) or \(x = 2\) and the solutions above..

Are there rational (non-integer) solutions beyond the integer ones?.

What methods/tools are useful to solve or prove completeness of solutions?.

Useful tools: modular arithmetic (local obstructions), substitution \(x = 2t\), factor/gcd analysis, infinite descent, Baker-type bounds, Chabauty–Coleman, and computer algebra systems (Sage, Magma) for rigorous verification.

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