Find rational a,b with \(\displaystyle{a}\ne{b}\) such that

Nevaeh Glass

Nevaeh Glass

Answered question

2022-04-13

Find rational a,b with ab such that x²bxa³b³3b=0 has rational solutions.

Answer & Explanation

CyncgotoCancey1k6

CyncgotoCancey1k6

Beginner2022-04-14Added 6 answers

By substituting x=12ab (or rather, a=bx12) and simplifying, the expression becomes
4b2124(x33122)
To make this a rational square, we need to find a rational point on the elliptic curve y2=x33122. The theory of rational points on elliptic curves is huge and complicated, so it is easiest to just let a computer do it.
According to Magma, the Mordell-Weil group (the group of rational points) on this curve is isomorphic to Z3, so it has only three rational points. Those points are just the ones we already expect, namely two points with x=12 (corresponding to a=b), and a point at infinity. So there are no other rational points.

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