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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 228.10028

**Autor: ** Erdös, Paul; Selfridge, J.L.

**Title: ** Some problems on the prime factors of consecutive integers. II. (In English)

**Source: ** Proc. Washington State Univ. Conf. Number Theory 1971, 13-21 (1971).

**Review: ** [For the entire collection see Zbl 225.00004.]

[Part I, Ill. J. Math. 11, 428-430 (1967; Zbl 149.28901).]

The authors develop a number of interesting results centering round the following interesting conjecture of C. A. Grimm: Let n+1, ... ,n+k be consecutive composite numbers. Then for each i,1 \leq i \leq k there is a p_{i}, p_{i} | n+i, p_{i1} \ne p_{i2} for i_{1} \ne i_{2}. Grimm also stated the following weaker conjecture: The product of k consecutive composite numbers need to have at least k prime factors. The interest in Grimm's conjectures is that even the weaker conjecture is enough to imply p_{i+1}-p_{i} << (p_{i}/ log p_{i})^{ ½}. Actually in view of a result of the reviewer the weaker conjecture implies that p_{i+1}-p_{i} << p^{ ½-c1}_{i} where c_{1} is a certain positive constant. This is known to the authors. These results show that there is not much hope to prove Grimm's conjectures in the ``near future''.

The authors prove a number of interesting results independent of any hypothesis. Let \nu(n,k) be the number of distinct prime factors of (n+1) ... (n+k); f_{1}(k) be the smallest integer k so that for every 1 \leq l \leq k, \nu (n,1) \geq 1 but \nu(n,k+1) = k; f_{0}(n) be the largest integer k for which \nu(n,k) \geq k. Let f_{2}(n) be the largest integer k so that for each 1 \leq i \leq k there is a p_{i} | n+i, p_{i1} \ne p_{i2} if i_{1} \ne i_{2}. Let p(m) be the greatest prime factor of m; f_{3}(n) the largest integer so that all the primes p(n+i), 1 \leq i \leq k, are distinct; f_{4}(n) be the largest integer k so that p(n+i) \geq i, 1 \leq i \leq k and f_{5}(n) be the largest integer k so that p(n+i) \geq k for every 1 \leq i \leq k. The main object of the paper is the study of the functions f_{i}(n), 0 \leq i \leq 5. We content by stating two theorems on f_{2}(n).

Theorem 3: f_{2}(n) > (1+0(1)) log n (this theorem has been improved by the reviewer to f_{2}(n) >> log n (log_{2} n/ log_{3}n)^{ ½}. Recently the reviewer received a letter from Tijdeman who says that he can improve this further to f_{2}(n) >> (log n)^{2}(log_{2}n)^{-8}; log_{r}n denotes the r-th iterated logarithm). Theorem (stated without proof): f_{2}(n) < \exp(c log n log_{3} n/ log_{2}n) for infinitely many n.

**Reviewer: ** K.Ramachandra

**Classif.: ** * 11N99 Multiplicative number theory

11N56 Rate of growth of arithmetic functions

**Citations: ** Zbl 149.289; Zbl 225.00004(EA)

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