Positive integer solutions to $3x^4-3x^2+1=y^2$

Asked Jan 26 '21 at 09:07

Active Jan 26 '21 at 09:07

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I need to solve $$3x^4-3x^2+1=y^2$$ over the positive integers.

I've tried setting $x^2=t$ and considering the discriminant. I've tried rewriting into $3(2x^2-1)^2+1=(2y)^2$. I've tried substituting $x=3a\pm 1$. The problem would be easy by bounding if the leading coefficient is a perfect square.

asked Jan 26 '21 at 09:07

Jolenia

Note: $y $ is even and $y=3m\pm1$ for some $m\in\mathbb {Z}^+$. – ultralegend5385 Jan 26 '21 at 09:35
See this post how to write it as elliptic curve (with $-3x^2$ instead of $3x^2$). – Dietrich Burde Jan 26 '21 at 09:37
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$y$ is always odd regardless of $x$ – Jolenia Jan 26 '21 at 09:42
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Also Art of Problem solving has several pages with solutions. Here the notation is $3p^4-3p^2+1=q^2$. Again here with $3q^4-3q^2+1=t^2$. – Dietrich Burde Jan 26 '21 at 09:47

Positive integer solutions to $3x^4-3x^2+1=y^2$

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