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**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 a_{1} < ... < a_{k} \leq n be asequence of integers for which all the sums a_{1}+a_{j} 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 a_{1}+a_{j} 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 p_{1} < p_{2} < ... 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, p_{n+1}-p_{n} > cL_{n} where L_{n} = (log n)(log log n)(log log log log n)/(log log log n)^{2}. The author wishes to see a proof that p_{n+1}-p_{n} > cL_{n} holds for every c. He has shown [Publ. Math. 1, 33-37 (1949; Zbl 033.16302)] that there is a constant c_{1} such that, for infinitely many n, **max** (p_{n+1}-p_{n},p_{n}-p_{n-1}) > c_{1}L_{n}, and *H.Maier* [Adv. Math. 39, 257-269 (1981; Zbl 457.10023)] has proved that for every k there is a constant c_{k} such that, for infinitely many n, **max**_{i = 1,2,...,k}(p_{n+k+1}-p_{n+1} > c_{k} L_{n}). Nevertheless, it is conjectured that **lim**_{x ––> oo} D_{k+1}(x)/D_{k}(x) = 0 where D_{k}(x) = **max**_{pn < x}**max**_{i = 1,...,k-1}(p_{n+i+1}-p_{n+i}). The list of miscellaneous problems begins with the problem of determining whether or not almost all integers have two divisors d_{1} and d_{2} satisfying d_{1} < d_{2} < 2d_{1}. The selection here is quite varied.

**Reviewer: ** B.Garrison

**Classif.: ** * 11-02 Research monographs (number theory)

11B13 Additive bases

11N05 Distribution of primes

00A07 Problem books

11P99 Additive number theory

11B99 Sequences and sets

11M99 Analytic theory of zeta and L-functions

**Keywords: ** problems on additive number theory; problems on prime numbers; miscellaneous problems; sequence of integers; distinct sums; difference of consecutive primes; divisors of integers

**Citations: ** Zbl.476.00003; Zbl.061.073; Zbl.019.394; Zbl.033.163; Zbl.457.10023

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag