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 xz and yz, 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 xmynzp has only solutions of Type 1 or Type 2, unless x2zkzp for some integer k ≥ 2 and non-empty prefix zp of z, or z2xlxp for some integer l ≥ 2 and non-empty prefix xp of x.

The program takes as input three partial words x, y, z and outputs values for m, n and p if such values exist to satisfy the equation.

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Acknowledgement:   This material is based upon work supported by the National Science Foundation under Grant No. DMS-0452020.

Disclaimer:   Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.