ON THE PRIME FACTORS OF \k/ AND OF CONSECUTIVE INTEGERS
نویسندگان
چکیده
1. First we introduce some notations. elements of the finite set S is denoted by ISI. We write ex = exp x. We denote the i th prime number by p i ,. v(n) denotes the number of distinct prime factors of n. mk is the smallest integer m for which v(1k» > k. P(n) is the greatest, p(n) the smallest prime factor of n. nk is the smallest integer n for which P(n+i) > k for all 1 < i <_ k. Mk > c I k 2 log k. (1) We prove THEOREM 1. Por k > k 0 we have (2) m k > c 2 k 2 (log k) 4/3 (log log k)-4/3(logloglog k)-1/3 It seems certain that m k > k2+c. It follows from results of Selfridge and the first author [41 that, by an averaging process, mk < ke+e The exact determination of, or even good inequalities for, mk seems very difficult. Denote by mk the least integer such that for every m ? mk k)I ? k Szemerédi and the first author proved in L51 that mk < (e+c) k for every c > 0 if k > k 0. Very likely mk < k c but we could not even prove mk < (e-F) k. Schinzel conjectured that v((k)) = k holds for
منابع مشابه
ACTÁ ARITHMETICA XLIX (1988) Prime factors of binomial coefficients and related problems
1 . Sequences of positive integers with the consecutive integer property. Consider a sequence of k positive integers {ai} = a,, . . ., a k with the following properties : (i) ai < k for all i, (ü) (the consecutive integer property) There is an n such that ai is the quotient when n + i is cleared of all its prime factors greater than k . Or, for each prime p k, the pattern of that prime and its ...
متن کاملOn the Maximal Length of Two Sequences of Consecutive Integers with the Same Prime Divisors
The initial motivation of the present paper is the following problem of Erd6s and Woods, whose solution would be of interest in logic (see [3, 6, 11, 10, 16]): Does there exist an integer k >=2 with the following property : i f x and y are positive integers such that for l<=i<=k, the two numbers x+ i and y+i have the same prime factors, then x=y. For each integer n->2, let us denote by Supp (n)...
متن کاملThe power digraphs of safe primes
A power digraph, denoted by $G(n,k)$, is a directed graph with $Z_{n}={0,1,..., n-1}$ as the set of vertices and $L={(x,y):x^{k}equiv y~(bmod , n)}$ as the edge set, where $n$ and $k$ are any positive integers. In this paper, the structure of $G(2q+1,k)$, where $q$ is a Sophie Germain prime is investigated. The primality tests for the integers of the form $n=2q+1$ are established in terms of th...
متن کاملOn the Number of Prime Factors of a Finite
contributions to Catalan's classical problem to show that 8,9 is the only pair of consecutive positive integers which are both perfect powers. The work of Professor Ko and the related work by others show that there is just one mathematical world and that there are no borders in this world. For any positive integer n let P (n); !(n); (n) denote the greatest prime factor of n, the number of disti...
متن کاملPrime factors of consecutive integers
This note contains a new algorithm for computing a function f(k) introduced by Erdős to measure the minimal gap size in the sequence of integers at least one of whose prime factors exceeds k. This algorithm enables us to show that f(k) is not monotone, verifying a conjecture of Ecklund and Eggleton.
متن کاملAsymptotic behaviour of associated primes of monomial ideals with combinatorial applications
Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$. We say that $I$ satisfies the persistence property if $mathrm{Ass}_R(R/I^k)subseteq mathrm{Ass}_R(R/I^{k+1})$ for all positive integers $kgeq 1$, which $mathrm{Ass}_R(R/I)$ denotes the set of associated prime ideals of $I$. In this paper, we introduce a class of square-free monomial ideals in the polynomial ring $R=K[x_1,ld...
متن کامل