## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  214.30603
Autor:  Erdös, Paul
Title:  Some applications of graph theory to number theory (In English)
Source:  Proc. 2nd Chapel Hill Conf. Combin. Math. Appl., Univ. North Carolina 1970, 136-145 (1970).
Review:  [For the entire collection see Zbl 208.00201.]
Several applications of graph theory to number theory are discussed mostly without proofs. The following result is proved in detail: Let a1 < ... < ak \leq x, k > \pi (x) be a sequence of integers. Denote by f(k,x) the smallest integer r so that there always are r primes p1, ... ,pr for which more than r a's are of the form prod ri = 1p\alphaii. The author and Straus proved

f(\pi (x)+1,x) = (4+o(1)){x ½ \over log x}.

A good estimate is given for f(k,x) if k = cx.
Classif.:  * 11B75 Combinatorial number theory
11N99 Multiplicative number theory
Citations:  Zbl 208.00201(EA)

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