## 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) a1 a2··· ak = z2, a1, a2,..., ak in A, a1 < a2 < ··· < ak, 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

Fk(n): = maxA\subseteq [1,n]; (1) is not solvable for A| A|, and
Lk(n): = maxA\subseteq [1,n]; (1) is not solvable for A suma in A 1/a

then for all n in N, F2(n) is equal to the number of square-free integers not exceeding n, that is F2(n) ~ 6/\pi2· n. Non-trivial upper and lower bounds for Fk(n) for k = 3,4,4j, 4j+2 are established and it is shown that (log 2-\epsilon)n < F2k+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 Lk are established for n ––> oo:
L4j = (1+o(1)) log log n, L4j+2 = (3/2+o(1)) log log n, L2j+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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