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On divisors of sums of integers. I. (English) Zbl 0612.10042
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.
Reviewer: K.Ramachandra

##### MSC:
 11P32 Goldbach-type theorems; other additive questions involving primes 11N05 Distribution of primes
##### Keywords:
non-empty subsets; cardinality; greatest prime factor
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##### References:
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