On Multiplicative Representations of Integers
نویسنده
چکیده
Let I < a,< . . . < ak < x ; b, < • • < b, < x . Assume that the number of solutions of a, b, = m is less than c. The authors prove that then (1) ki< ló x(log log x)"`' g They also give a simple proof of Szemerédi's theorem : If the products a ;b; are all distinct then (2) ki < c2x (i .e . f(1)=0) . log x They conjecture that (2) holds for c_ = 1 + E if x > x o(e ) . Several other unsolved problems are stated . Let a, < . . . < ak < x be a sequence of integers for which the products a ;a, are all distinct . Erdős proved that '77'(x ) + c2x 314/(log x ) 312 < max k < -;r (x) + c,x "'/(log x)"' . Perhaps there is an absolute constant c so that x3" (1) max k = ;r (x ) + cx 3 "/ (1og x )3'2 + o, ((1OX)3/2 ) g but we can not prove (1) . (c, c,, • denote absolute constants not necessarily the same.) Erdős (1964a) also proved that if a, < . . . < ak ~ x is such that the number of solutions of a;aj = t is less than 2' + 1 then (2) max k = (1 + Q(1)) n(log logn)` ' (1 1)! log n In fact (2) holds if the number of solutions is < 2' ' + 1 . Let a, < • • •, denote by g(n) the number of solutions of n = a;a1. (2) easily
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