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In the post Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$, the OP presents a certain method of solution, and asks for advice.

While perusing it, I had a brainstorm for a very different method of attack. But I'm stuck half-way through my own solution, and want to get some help. Should I start a new, linked post (to acknowledge the different method being used), or post an answer or comment in the existing thread (to acknowledge that I'm trying to solve the same equation)?

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From your brief description, it sounds like you do indeed have your own independent question. It may be worthwhile linking to the other topic as a reference. Be careful to frame your question as "I want to understand how to make an argument like this work" rather than "Tell me how to find solutions to this equation".

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