BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//CS-TR-75-485
       ENTRY:: August 23, 1995
ORGANIZATION:: Stanford University, Department of Computer Science
       TITLE:: On multiplicative representations of integers.
        TYPE:: Technical Report
      AUTHOR:: Erdoes, Paul
      AUTHOR:: Szemeredi, Endre
        DATE:: March 1975
       PAGES:: 19
    ABSTRACT:: In 1969 it was shown by P. Erdoes that if 0 < $a_1$ < $a_2$ <
               ... < $a_k \leq x$ is a sequence of integers for which the
               products $a_i a_j$ are all distinct then the maximum possible
               value of k satisfies

               $\pi$(x) + $c_2$ $x^{3/4}$/${(log x)}^{3/2}$ < max k <
               $\pi$(x) + $c_1$ $x^{3/4}$/$(log x)^{3/2}$

               where $\pi$(x) denotes the number of primes not exceeding x
               and $c_1$ and $c_2$ are absolute constants.

               In this paper we will be concerned with similar results of
               the following type. Suppose 0 < $a_1$ < ... < $a_k \leq x$, 0
               < $b_1$ < ... < $b_{\ell} \leq x$ are sequences of integers.
               Let g(n) denote the number of representations of n in the
               form $a_i b_j$. Then we prove:

               (i) If g(n) $\leq$ 1 for all n then for some constant $c_3$,

               k$\ell$ < $c_3 x^2$/log x.

               (ii) For every c there is an f(c) so that if g(n) $\leq$ c
               for all n then for some constant $c_4$,

               k$\ell$ < $c_4 x^2$/log x ${(log log x)}^{f(c)}.
       NOTES:: [Adminitrivia V1/Prg/19950823]
         END:: STAN//CS-TR-75-485