Let $$2^{1/3}=a_0+\frac{1}{a_1+\dots}$$ be the canonical continued fraction of $2^{1/3}$. It is relatively easy to prove that if $a_n<10\phi^n$ then the equation $x^3-2y^3=1$ has no nontrivial integer solutions (this equation has no solutions anyway, but it would be an interesting alternative, and relatively easy, proof). The first few terms actually are:
$2^{1/3}= $[1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, 4, 1, 1, 2, 14, 3, 12, 1, 15, 3, 1, 4, 534, 1, 1, 5, 1, 1, 121, 1, 2, 2, 4, 10, 3, 2, 2, 41, 1, 1, 1, 3, 7, 2, 2, 9, 4, 1, 3, 7, 6, 1, 1, 2, 2, 9, 3, 1, 1, 69, 4, 4, 5, 12, 1, 1, 5, 15, 1, 4, 1, 1, 1, 1, 1, 89, 1, 22, 186, 6, 2, 3, 1, 3, 2, 1, 1, 5, 1, 3, 1, 8, 9, 1, 26, 1, 7, 1, 18, 6, 1, 372, 3, 13, 1, 1, 14, 2, 2, 2, 1, 1, 4, 3, 2, 2, 1, 1, 9, 1, 6, 1, 38, 1, 2, 25, 1, 4, 2, 44, 1, 22, 2, 12, 11, 1, 1, 49, 2, 6, 8, 2, 3, 2, 1, 3, 5, 1, 1, 1, 3, 1, 2, 1, 2, 4, 1, 1, 3, 2, 1, 9, 4, 1, 4, 1, 2, 1, 27, 1, 1, 5, 5, 1, 3, 2, 1, 2, 2, 3, 1, 4, 2, 2, 8, 4, 1, 6, 1, 1, 1, 36, 9, 13, 9, 3, 6, 2, 5, 1, 1, 1, 2, 10, 21, 1, 1, 1, 2, 1, 2, 6, 2, 1, 6, 19, 1, 1, 18, 1, 2, 1, 1, 1, 27, 1, 1, 10, 3, 11, 38, 7, 1, 1, 1, 3, 1, 8, 1, 5, 1, 5, 4, 4, 4, 7, 2, 1, 21, 1, 1, 5, 10, 3, 1, 72, 6, 9, 1, 3, 3, 2, 1, 4, 2, 1, 1, 1, 1, 2, 1, 7, 8, 1, 2, 1, 8, 1, 8, 3, 1, 1, 3, 2, 1, 8, 1, 1, 1, 1, 1, 6, 1, 4, 3, 4, 1, 1, 1, 4, 30, 39, 2, 1, 3, 8, 1, 1, 2, 1, 3, 1, 9, 1, 4, 1, 2, 2, 1, 6, 2, 1, 1, 3, 1, 4, 1, 2, 1, 1, 5, 1, 2, 10, 1, 5, 4, 1, 1, 4, 1, 2, 1, 1, 2, 12, 2, 1, 8, 3, 2, 6, 1, 3, 10, 1, 2, 20, 1, 6, 1, 2, 186, 2, 2, 1, 2, 47, 1, 19, 2, 2, 1, 1, 1, 2, 1, 1, 3, 2, 8, 1, 18, 3, 5, 39, 1, 2, 1, 1, 1, 1, 4, 1, 5, 2, 6, 3, 1, 1, 1, 4, 2, 1, 6, 1, 1, 220, 1, 3, 1, 3, 1, 4, 5, 1, 2, 1, 13, 2, 2, 2, 1, 1, 1, 1, 7, 2, 1, 7, 1, 3, 1, 1, 11, 1, 2, 2, 4, 2, 33, 3, 1, 1, 2, 6, 3, 1, 1, 3, 6, 8, 3, 4, 84, 1, 1, 2, 1, 10, 2, 2, 20, 1, 3, 1, 7, 13, 14, 1, 29, 1, 1, 5, 1, 7, 1, 1, 2, 1, 56, 1, 3, 2, 1, 13, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 255, 2, 4, 5, 1, 1, 1, 3, 1, 3, 3, 1, 6, 1, 1, 6, 1, 71, 1, 9, 1, 2, 1, 11, 5, 1, 25, 1, 6, 67, 2, 9, 6, 1, 5, 2, 15, 1, 2, 48, 2, 7, 1, 3, 1, 4, 21, 1, 1, 2, 1, 27, 3, 26, 2, 1, 1, 2, 5, 7, 3, 7451, 2, 29, 4, 3, 8, 17, 3, 8, 2, 3, 1, 1, 1, 5, 113, 1, 3, 4, 1, 4, 1, 1, 13, 1, 34, 1, 2, 7, 1, 3, 3, 7, 1, 3, 1, 1, 4, 2, 69, 1, 3, 12, 34, 1, 2, 151, 1, 4941, 4, 1, 1, 12, 3, 4, 2, 3, 1, 1, 1, 1, 1, 2, 1, 1, 6, 16, 1, 2, 27, 2, 13, 4, 1, 1, 1, 3, 11, 1, 1, 3, 1, 53, 2, 15, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 1, 9, 1, 1, 10, 3, 1, 1, 2, 1, 2, 2, 1, 10, 9, 1, 2, 5, 1, 2, 2, 1, 1, 2, 4, 7, 1, 5, 1, 1, 1, 1, 4, 2, 25, 16, 5, 4, 1, 3, 2, 3, 13, 1, 49, 6, 2, 5, 1, 1, 2, 7, 3, 2, 1, 1, 1, 4, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 4, 1, 2, 1, 10, 5, 4, 8, 10, 2, 4, 1, 1, 1, 4, 1, 41, 1, 3, 1, 56, 3, 1, 1, 3, 1, 3, 1, 5, 6, 6, 3, 1, 2, 1, 1, 1, 12, 1, 10, 2, 1, 1, 1, 1, 50, 5, 1, 2, 6, 5, 1, 2, 5, 6, 5, 2, 77, 1, 4, 2, 1, 1, 1, 1, 1, 4, 2, 1, 2, 1, 1, 1, 1, 1, 6, 2, 1, 1, 7, 1, 5, 1, 1, 1, 1, 2, 2, 1, 1, 5, 2, 1, 5, 1, 1, 1, 4, 1, 2, 17, 1, 20, 7, 4, 2, 1, 1, 1, 2, 1, 4, 7, 3, 4, 3, 3, 5, 31, 1, 1, 2, 2, 6, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 6, 1, 1, 23, 20, 1, 22, 16, 4, 2, 1, 3, 2, 1, 1, 2, 5, 5, 1, 1, 15, 3, 1, 1, 2, 1, 1, 1, 4, 2, 1, 2, 23, 6, 10, 3, 2, 3, 6, 2, 1, 1, 1, 1, 1, 1, 4, 3, 2, 1, 2, 1, 4, 10, 7, 1, 1, 1, 1, 3, 3, 2, 108, 1, 1, 11, 2, 6, 1, 4, 1, 2, 2, 9, 3, 1, 1, 3, 22, 4, 1, 93, 1, 3, 1, 4, 2, 1, 2, 3, 2, 1, 2, 11, 1, 1, 3, 1, 2, 1, 28, 23, 4, 11, 1, 9, 1, 4, 3, 1, 6, 1, 2, 1, 12, 2, 6, 19, 1, 4, 4, 2, 1, 1, 1, 1, 1, 10, 3, 4, 5, 4, 4, 1, 43, 1, 1, 5, 1, 73, 6, 35, 5, 4, 1, 13, 2, 1, 6, 1, 4, 2, 1, 1, 1, 27, 1, 1, 4, 1, 1, 7, 1, 1, 14, 9, 14, 1, 1, 1, 1, 10, 3, 3, 1, 2, 5, 7, 4, 2, 6, 4, 2, 4, 20, 1, 5, 2, 1, 1, 3, 1, 1, 2, 1, 2, 2, 11, 1, 1, 14, 1, 1, 14, 1, 9, 1, 1, 1, 33, 2, 1, 3, 7, 2, 3, 1, 2, 1, 27, 1, 1, 2, 34, 2, 1, 8, 6, 4, 1, 1, 7, 1, 5, 3, 1, 354, 3, 2, 1, 1, 2, 6, 1, 1, 3, 17, 1, 3, 1, 195, 1, 6, 1, 1, 1, 25, 1, 3, 1, 2, 304, 5, 1, 1, 5, 1, 2, 9, 4, 1, 1, 4, 2, 1, 44, 1, 1, 1, 2, 8, 16, 1, 204, 1, 1, 1, 2, 1, 2, 3, 3, 1, 1, 20, 1, 11, 1, 8, 1, 2, 16, 1, 1, 1, 10, 5, 1, 10, 6, 1, 5, 2, 1, 33, 6, 6, 1, 3, 1, 1, 28, 1, 10, 3, 1, 1, 23, 2, 2, 1, 7, 4, 1, 4, 1, 7, 1, 2, 1, 1, 1, 2, 2, 2, 7, 1, 5, 201, 6, 1, 10, 8, 9, 1, 3, 3, 8, 5, 1, 136, 1, 2, 6, 2, 1, 2, 2, 9, 1, 4, 2, 2, 5, 1, 29, 1, 1, 4, 12, 1, 6, 4, 1, 13, 1, 2, 1, 2, 6, 127, 2, 2, 1, 1, 119, 1, 124, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 10, 1, 9, 3, 2, 3, 2, 4, 1, 8, 1, 2, 2, 2, 3, 1, 8, 1, 4, 3, 2, 8, 3, 1, 3, 1, 6, 3, 2, 4, 1, 3, 4, 1, 1, 5, 1, 4, 4, 9, 2, 3, 16, 1, 1, 2, 1, 3, 16, 1, 4, 2, 1, 10, 1, 4, 1, 2, 6, 4, 1, 5, 6, 2, 1, 1, 1, 6, 1, 1, 11, 1, 2, 23, 1, 7, 11, 2, 2, 1, 4, 1, 2, 1, 2, 49, 8, 1, 3, 3, 1, 1, 3, 2, 2, 2, 1, 1, 1, 1, 4, 2, 1, 4, 1, 2, 8, 1, 9, 3, 9, 1, 1, 5, 23, 5, 11, 1, 1, 1, 1, 1, 90, 2, 1, 5, 56, 1, 2, 3, 1, 8, 85, 1, 1, 131, 2, 1, 4, 2, 1, 1, 37, 1, 14, 1, 5, 2, 4, 1, 1, 1, 11, 1, 1, 1, 2, 2, 1, 1, 1, 52, 18, 4, 7, 2, 101, 2, 1, 119, 1, 1, 1, 3, 1, 13, 3, 1, 1, 6, 1, 1, 27, 2, 1, 2, 3, 3, 4, 3, 11, 1, 5, 3, 1, 9, 1, 1, 15, 1, 1, 2, 1, 2, 4, 15, 2, 1, 2, 7, 1, 5, 1, 6, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 2, 23, 1, 1, 19, 1, 5, 12, 3, 5, 1, 1, 10, 2, 1, 3, 1, 2, 3, 3, 1, 22, 1, 2, 1, 1, 1, 1, 1, 5, 1, 2, 36, 2, 1, 1, 1, 4, 3, 10, 1, 14, 6, 1, 5, 2, 1, 328, 1, 1, 2, 1, 1, 2, 11, 1, 2, 1, 1, 1, 7, 1, 2, 16, 2...]
More here.
So if we have an even weak upper bound on the coefficients on the continued fraction of an algebraic number we can solve lots of diophantine equations using approximation arguments. I think even something ridiculous like $$\alpha=[a_0;a_1\dots]\implies a_n < e^{e^{e^n}}$$ for some known algebraic $\alpha$ might prove useful. It looks like such a statement should be easy to prove but unfortunately I don't have anything so far.