Large solutions [to Diophantine equations]

A few other 2-variable equations with fairly large smallest solutions are:

• x^^{3} == 3*y^^{3} -x y + 63: {7149, 4957}

• x^^{4} == y^^{3} + 2 x y - 2 y + 81: {19674, 531117}

• x^^{4} == 5*y^^{3} + x y + y - 8 x: {69126, 1659072}

The equation x^^{x} y^^{y} == z^^{z} is known to have smallest non-trivial solution {2985984, 1679616, 4478976}.