Cambridge Entrance Exam: Can You Solve This Diophantine Equation?

Find all integers m and n such that m^3=n^3+2n^2-1

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And, as always, any comments, feedback or suggestions are welcomed!

Thanks.

[ademkesaev]

Thank you for sharing! I plan on doing the STEP exam next year and I enjoy the questions because they are just the right amount of challenging, not too hard but make me have to think a lot

[subashgurung]

Is this problem related to number theory or algebra?