منابع مشابه
Sums of Prime Divisors and Mersenne Numbers
The study of the function β(n) originated in the paper of Nelson, Penney, and Pomerance [7], where the question was raised as to whether the set of Ruth-Aaron numbers (i.e., natural numbers n for which β(n) = β(n+ 1)) has zero density in the set of all positive integers. This question was answered in the affirmative by Erdős and Pomerance [5], and the main result of [5] was later improved by Po...
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Let Bn (n = 0, 1, 2, . . .) denote the usual n-th Bernoulli number. Let l be a positive even integer where l = 12 or l ≥ 16. It is well known that the numerator of the reduced quotient Bl/l is a product of powers of irregular primes. Let (p, l) be an irregular pair with p below 12 million. We show that for every r ≥ 1 the congruence Bmr/mr ≡ 0 (mod p ) has a unique solution for mr where l ≡ mr ...
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Let π(x; d, a) denote the number of primes p ≤ x with p ≡ a(mod d). Chebyshev’s bias is the phenomenon for which “more often” π(x; d, n) > π(x; d, r), than the other way around, where n is a quadratic nonresidue mod d and r is a quadratic residue mod d. If π(x; d, n) ≥ π(x; d, r) for every x up to some large number, then one expects that N(x; d, n) ≥ N(x; d, r) for every x. Here N(x; d, a) deno...
متن کاملPrime 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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Let d(m) be the number of divisors of the positive integer m. Here, we show that if n 6∈ {3, 5}, then d(n!) is a divisor of n!. We also show that the only positive integers n such that d(Fn) divides Fn, where Fn is the nth Fibonacci number, are n ∈ {1, 2, 3, 6, 24, 48}.
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ژورنال
عنوان ژورنال: Pacific Journal of Mathematics
سال: 1961
ISSN: 0030-8730,0030-8730
DOI: 10.2140/pjm.1961.11.379