A lower-bound conjecture for nonzero differences of monomials

Let ${\mathbf a},{\mathbf b},{\mathbf c}\to be fixed positive integers, let$ _1, _2, _3\to be positive integers, and let $\varepsilon>0$. Define

$$D=n_1^a-n_2^b n_3^c.$$

Monomial-difference lower-bound conjecture. If $D e 0$, then

$$|D|\gg n_1^{a-1-\varepsilon}n_2^{-1}n_3^{-1},$$

where the implicit constant depends only on $\varepsilon$.

This is introduced as a weak form of the $abc$ conjecture and is intended to sharpen error terms in the integral involving three zeta values when the coefficients are integers. Its resolution status is not specified in the supplied text.