We solve two long-standing open problems on word equations. Firstly, we prove that a onevariable word equation with constants has either at most three or an infinite number of solutions. The existence of such a bound had been conjectured, and the bound three is optimal. Secondly, we consider independent systems of three-variable word equations without constants. If such a system has a nonperiodic solution, then this system of equations is at most of size 17. Although probably not optimal, this

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... s the first finite bound found. However, the conjecture of that bound being actually two still remains open.