# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  A. Balog and A. Sárközy,On sums of sequences of integers, II, to appear.  H. Davenport, On the addition of residue classes,J. London Math. Soc.,10 (1935), 30–32. · Zbl 0010.38905  H. Davenport, A historical note,J. London Math. Soc.,22 (1947), 100–101. · Zbl 0029.34401  P. Erdos and P. Turán, On a problem in the elementary theory of numbers,American Math. Monthly,41 (1934), 608–611. · Zbl 0010.29401  P. X. Gallagher, A larger sieve,Acta Arith.,18 (1971), 77–81. · Zbl 0231.10028  G. H. Hardy and E. M. Wright,An introduction to the theory of numbers, 5th ed. (Oxford, 1979). · Zbl 0423.10001  G. Pólya and G. Szego,Problems and theorems in analysis, Vol. I, Springer-Verlag, (Berlin, 1972).  H. E. Richert,Lectures on sieve methods, Tata Institute of Fundamental Research (Bombay, 1976). · Zbl 0392.10041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.