منابع مشابه
Prime divisors of palindromes
Abstract In this paper, we study some divisibility properties of palindromic numbers in a fixed base g ≥ 2. In particular, if PL denotes the set of palindromes with precisely L digits, we show that for any sufficiently large value of L there exists a palindrome n ∈ PL with at least (log log n)1+o(1) distinct prime divisors, and there exists a palindrome n ∈ PL with a prime factor of size at lea...
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We show that recent results of Coppersmith, Boneh, Durfee and Howgrave-Graham actually apply in the more general setting of (partially) approximate common divisors. This leads us to consider the question of “fully” approximate common divisors, i.e. where both integers are only known by approximations. We explain the lattice techniques in both the partial and general cases. As an application of ...
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The Liouville function is defined by λ(n) := (−1)Ω(n) where Ω(n) is the number of prime divisors of n counting multiplicity. Let ζm := e2πi/m be a primitive m–th root of unity. As a generalization of Liouville’s function, we study the functions λm,k(n) := ζ kΩ(n) m . Using properties of these functions, we give a weak equidistribution result for Ω(n) among residue classes. More formally, we sho...
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We define the arithmetic function P by P (1) = 0, and P (n) = p1 + p2 + · · ·+ pk if n has the unique prime factorization given by n = ∏k i=1 p ai i ; we also define ω(n) = k and ω(1) = 0. We study pairs (n, n+ 1) of consecutive integers such that P (n) = P (n+ 1). We prove that (5, 6), (24, 25), and (49, 50) are the only such pairs (n, n + 1) where {ω(n), ω(n + 1)} = {1, 2}. We also show how t...
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We study the values of arithmetic functions taken on the elements of a non-homogeneous Beatty sequence αn+ β , n= 1,2, . . . , where α,β ∈R, and α > 0 is irrational. For example, we show that ∑ n N ω ( αn+ β )∼N log logN and ∑ n N (−1)Ω( αn+β ) = o(N), where Ω(k) and ω(k) denote the number of prime divisors of an integer k = 0 counted with and without multiplicities, respectively. © 2006 Elsevi...
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ژورنال
عنوان ژورنال: The Annals of Probability
سال: 1984
ISSN: 0091-1798
DOI: 10.1214/aop/1176993149