Modular Arithmetic Contradiction Trick for Diophantine Equations

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Diophantine equations (for us, these are equations where we are seeking only integer solutions) sometimes have no solutions. In such cases, it might be possible to reduce the polynomial equation to a certain modulus and show that no combination of residues for the the variables produces the residue on the other side. There is no official name for this method, but we call it the "modular arithmetic contradiction trick." While there is no known method for finding a suitable such modulus, we provide some heuristics in this video, for example involving Sophie Germain primes. These heuristics can be useful on math contests and olympiads.

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Can we solve
a³-b²=2
In the same way?

shafikbara