x^myⁿ ↑ z^p

This equation's solutions can be classified into the following categories, or solved according to the Theorem:

If a partial word w exists such that x, y, and z are contained in powers of w, then the solution is Trivial, or Type 1.

If the partial words x, y, and z satisfy x ↑ z and y ↑ z, then the solution is a Type 2 solution. Additionally, if z is a full word, then it is a Trivial solution.

Theorem: Let x, y, and z be primitive partial words such that (x, z) and (y, z) are good pairs. Let m, n, and p be integers such that m ≥ 2, n ≥ 2, and p ≥ 4. Then the equation x^myⁿ ↑ z^p has only solutions of Type 1 or Type 2, unless x² ↑ z^kz_p for some integer k ≥ 2 and non-empty prefix z_p of z, or z² ↑ x^lx_p for some integer l ≥ 2 and non-empty prefix x_p of x.