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Let F₊, ₃ (n) be the maximal size of a set A n such that the equation ₁a₂ aₖ=xᵈ, \; a₁<a₂<<aₖ\ has no solution with a₁, a₂, , aₖ A and integer x. Erdos, S\'ark\"ozy and T. S\'os studied F₊, ₂, and gave bounds when k=2, 3, 4, 6 and also in the general case. We study the problem for d=3, and provide bounds for k=2, 3, 4, 6 and 9, furthermore, in the general case, as well. In particular, we refute an 18 years old conjecture of Verstra\"ete. We also introduce another function f₊, ₃ closely related to F₊, ₃: While the original problem requires a₁, , aₖ to all be distinct, we can relax this and only require that the multiset of the aᵢ's cannot be partitioned into d-tuples where each d-tuple consists of d copies of the same number.
Fleiner et al. (Mon,) studied this question.