## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  484.10001
Autor:  Erdös, Paul
Title:  Some new problems and results in number theory. (In English)
Source:  Number theory, Proc. 3rd Matsci. Conf., Mysore/India 1981, Lect. Notes Math. 938, 50-74 (1982).
Review:  [For the entire collection see Zbl 476.00003.]
The problems discussed in this paper are ut into three groups: problems on additive number theory, problems on prime numbers, ans miscellaneous problems. Some of these have been solved, or partially solved, but no solutions are given here. Instead, the author directs our attention to intriguing questions remaining to be answered. He alse continues his custom of offering monetary rewards for solutions to some of the problems. For an example from the first group, let 1 \leq a1 < ... < ak \leq n be asequence of integers for which all the sums a1+aj are distinct. The author and P. Turán have shown [J. Lond. Math. Soc. 16, 212-215 (1941; Zbl 061.07301)] that max k = (1+o(1))n ½ and he conjectures that max k = n ½+O(1). If the hypothesis weakened so that the number of distinct a1+aj is (1+o(1))(k/2) it is no longer true that max k = (1+o(1))n ½ and an example exists with k \geq 2n ½/3 ½. Hoever, it is conjectured that max k = < cn ½ for some c < 2 ½. Now let p1 < p2 < ... be the sequence of consecutive primes. R.Rankin [J. Lond. Math. Soc. 13, 242-247 (1938; Zbl 019.39403)] has shown that fot infinitely many n, and for some c > 0, pn+1-pn > cLn where

Ln = (log n)(log log n)(log log log log n)/(log log log n)2.

The author wishes to see a proof that pn+1-pn > cLn holds for every c. He has shown [Publ. Math. 1, 33-37 (1949; Zbl 033.16302)] that there is a constant c1 such that, for infinitely many n, max (pn+1-pn,pn-pn-1) > c1Ln, and H.Maier [Adv. Math. 39, 257-269 (1981; Zbl 457.10023)] has proved that for every k there is a constant ck such that, for infinitely many n, maxi = 1,2,...,k(pn+k+1-pn+1 > ck Ln). Nevertheless, it is conjectured that limx ––> oo Dk+1(x)/Dk(x) = 0 where Dk(x) = maxpn < xmaxi = 1,...,k-1(pn+i+1-pn+i). The list of miscellaneous problems begins with the problem of determining whether or not almost all integers have two divisors d1 and d2 satisfying d1 < d2 < 2d1. The selection here is quite varied.
Reviewer:  B.Garrison
Classif.:  * 11-02 Research monographs (number theory)