Divisibility properties of integers $x,\ k$ satisfying $1\sp k+\cdots+(x-1)\sp k=x\sp k$
نویسندگان
چکیده
منابع مشابه
DIVISIBILITY PROPERTIES OF INTEGERS x , k SATISFYING lk + ■ ■ ■ + ( x - i ) k = xk
Based on congruences mod p and on properties of Bernoulli polynomials and Bernoulli numbers, several conditions are derived for x, k > 2 to satisfy the Diophantine equation 1k + 2k H-\(x 1 )* = xk . It is proved that ord2(x 3) = orà2k + 3 and that x cannot be divisible by any regular prime. Furthermore, by using the results of experiments with the above conditions on an SGI workstation it is pr...
متن کاملOn Divisibility Properties of Integers
1. Throughout this paper, we use the following notations : For any real number x let [x] denote the greatest integer less than or equal to x, and let lxll denote the distance from x to the nearest integer : jjxjl =min (x-[x], 1 + +[x]-x) . We write e""=e(x). The cardinality of the set X is denoted by JXJ . A (n) denotes the Mangoldt symbol . In this paper, our goal is to study the following pro...
متن کاملOn Divisibility Properties of Sequences of Integers
Let a 1 < a, < . . . be an infinite sequence of integers of positive lower logarithmic density, in other words 1 (1) lim sup > 0. X=+logxa;<x a i DAVENPORT and ERDŐS [1] proved that then there exists an infinite subsequence a,,, < a„, ` . . . satisfying a,, ./a,, .+, . In this note we will give various sharpenings of this result . The sequence a1 < a2 < . . . will be denoted by A, an infinite s...
متن کاملProperties of k-Rings and Rings Satisfying Similar conditions
Jacobson introduced the concept of K-rings, continuing the investigation of Kaplansky and Herstein into the commutativity of rings. In this note we focus on the ring-theoretic properties of K-rings. We first construct basic examples of K-rings to be handled easily. It is shown that a semiprime K-ring of bounded index of nilpotency is a commutative domain. It is proved that if R is a prime K-rin...
متن کاملErgodic Properties of k-Free Integers in Number Fields
Let K/Q be a degree d extension. Inside the ring of integers OK we define the set of k-free integers Fk and a natural OK-action on the space of binary OK-indexed sequences, equipped with an OK-invariant probability measure associated to Fk. We prove that this action is ergodic, has pure point spectrum, and is isomorphic to a Zdaction on a compact abelian group. In particular, it is not weakly m...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1994
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-1994-1257577-1