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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

11P32 Goldbach-type theorems; other additive questions involving primes
11N05 Distribution of primes
Full Text: DOI
[1] A. Balog and A. Sárközy,On sums of sequences of integers, II, to appear.
[2] H. Davenport, On the addition of residue classes,J. London Math. Soc.,10 (1935), 30–32. · Zbl 0010.38905
[3] H. Davenport, A historical note,J. London Math. Soc.,22 (1947), 100–101. · Zbl 0029.34401
[4] P. Erdos and P. Turán, On a problem in the elementary theory of numbers,American Math. Monthly,41 (1934), 608–611. · Zbl 0010.29401
[5] P. X. Gallagher, A larger sieve,Acta Arith.,18 (1971), 77–81. · Zbl 0231.10028
[6] G. H. Hardy and E. M. Wright,An introduction to the theory of numbers, 5th ed. (Oxford, 1979). · Zbl 0423.10001
[7] G. Pólya and G. Szego,Problems and theorems in analysis, Vol. I, Springer-Verlag, (Berlin, 1972).
[8] H. E. Richert,Lectures on sieve methods, Tata Institute of Fundamental Research (Bombay, 1976). · Zbl 0392.10041
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