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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 840.11010

**Autor: ** Erdös, Paul; Sárközy, A.; Sós, V.T.

**Title: ** On the product representation of powers. I. (In English)

**Source: ** Eur. J. Comb. 16, No.6, 567-588 (1995).

**Review: ** The authors study the solvability of the equation (1) a_{1} a_{2}··· a_{k} = z^{2}, a_{1}, a_{2},..., a_{k} in A, a_{1} < a_{2} < ··· < a_{k}, x in **N** for fixed k and dense sets A of natural numbers. It is shown that if k is even, k \geq 4, and if A is of positive density then the above equation is solvable. In particular, it is proved that if F_{k}(n): = **max**_{A\subseteq [1,n]; (1) is not solvable for A}| A|, and

L_{k}(n): = **max**_{A\subseteq [1,n]; (1) is not solvable for A} **sum**_{a in A} ^{1}/_{a} then for all n in **N**, F_{2}(n) is equal to the number of square-free integers not exceeding n, that is F_{2}(n) ~ 6/\pi^{2}· n. Non-trivial upper and lower bounds for F_{k}(n) for k = 3,4,4j, 4j+2 are established and it is shown that (log 2-\epsilon)n < F_{2k+1} < n-(1- \epsilon) n(log n)^{-2} (the authors remark that the above lower bound could be improved slightly by a lengthy computation). Moreover, the following results on L_{k} are established for n ––> oo:

L_{4j} = (1+o(1)) log log n, L_{4j+2} = (3/2+o(1)) log log n, L_{2j+1} = 1+(½+o(1)) log n.

**Reviewer: ** M.Helm (Stuttgart)

**Classif.: ** * 11B83 Special sequences of integers and polynomials

**Keywords: ** hybrid problems; product representation of squares

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