On divisors of sums of integers. I
Let N be a positive integer and let \(A_ 1,...,A_ k\) be non-empty subsets of \(\{\) 1,2,...,N\(\}\). Let \(| A_ i|\) denote the cardinality of \(A_ i\). For any integer larger than one let P(n) denote the greatest prime factor of n. The authors study \(P(a_ 1+a_ 2+...+a_ k)\) where \(a_ 1,...,a_ k\) are suitably chosen from the k sets \(A_ 1,...,A_ k\), respectively. They write \(T=(\prod^{k}_{j=1}| A_ j|)^{1/k}\). They prove the following interesting theorem: Let \(A_ 1,...,A_ k\) be non-empty subsets of \(\{\) 1,...,N\(\}\) with \(| A_ 1| =\min_{i}| | A_ i|\) and \(k>1\), and let \(\epsilon\) be a positive real number. If \(\sum^{k}_{i=1}| A_ i| >(1+\epsilon)N,\) then for any prime p with \(N<p<(1+\epsilon /2)N\), there exist \(a_ i\in A_ i\), for \(i=1,...,k\), such that \[ (1)\quad P(a_ 1+...+a_ k)=p\quad, \] whenever \(N>N_ 0(\epsilon,k)\). If \(T>8N^{1/2} \log N\), then there exist \(a_ i\in A_ i\), for \(i=1,...,k\), such that \[ (2)\quad P(a_ 1+...+a_ k)\quad >\quad kT/14 \log T\quad, \] for \(N>N_ 1(k)\). Further, there exist \(a_ i\in A_ i\), for \(i=1,...,k\), such that \[ (3)\quad P(a_ 1+...+a_ k)\quad >\quad | A_ 1| /N^{1/k+\epsilon}\quad, \] for \(N>N_ 2(\epsilon,k)\). Here \(N_ 0(\epsilon,k)\), \(N_ 1(k)\) and \(N_ 2(\epsilon,k)\) are numbers which are effectively computable in terms of \(\epsilon\) and k, k, and \(\epsilon\) and k, respectively.
scientific article; zbMATH DE number 3989470