Publications of (and about) Paul Erdös
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 pi, pi | n+i, pi1 \ne pi2 for i1 \ne i2. 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 pi+1-pi << (pi/ log pi) ½. Actually in view of a result of the reviewer the weaker conjecture implies that pi+1-pi << p ½-c1i where c1 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); f1(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; f0(n) be the largest integer k for which \nu(n,k) \geq k. Let f2(n) be the largest integer k so that for each 1 \leq i \leq k there is a pi | n+i, pi1 \ne pi2 if i1 \ne i2. Let p(m) be the greatest prime factor of m; f3(n) the largest integer so that all the primes p(n+i), 1 \leq i \leq k, are distinct; f4(n) be the largest integer k so that p(n+i) \geq i, 1 \leq i \leq k and f5(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 fi(n), 0 \leq i \leq 5. We content by stating two theorems on f2(n).
Theorem 3: f2(n) > (1+0(1)) log n (this theorem has been improved by the reviewer to f2(n) >> log n (log2 n/ log3n) ½. Recently the reviewer received a letter from Tijdeman who says that he can improve this further to f2(n) >> (log n)2(log2n)-8; logrn denotes the r-th iterated logarithm). Theorem (stated without proof): f2(n) < \exp(c log n log3 n/ log2n) for infinitely many n.
Classif.: * 11N99 Multiplicative number theory
11N56 Rate of growth of arithmetic functions
Citations: Zbl 149.289; Zbl 225.00004(EA)
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