ON FINITE SUMS OF RECIPROCALS OF DISTINCT nTR POWERS
نویسنده
چکیده
Introduction* It has long been known that every positive rational number can be represented as a finite sum of reciprocals of distinct positive integers (the first proof having been given by Leonardo Pisano [6] in 1202). It is the purpose of this paper to characterize {cf. Theorem 4) those rational numbers which can be written as finite sums of reciprocals of distinct nth. powers of integers, where n is an arbitrary (fixed) positive integer and "finite sum" denotes a sum with a finite number of summands. It will follow, for example, that p\q is the finite sum of reciprocals of distinct squares if and only if
منابع مشابه
Sums of Reciprocals of the Central Binomial Coefficients
We consider a set of combinatorial sums involving the reciprocals of the central binomial coefficients and try to solve (or close) them by means of generating functions. We obtain a number of results for infinite sums, in some of which the golden ratio φ appears. Besides, we close some finite sums by applying the method of coefficients to the generating functions previously obtained.
متن کاملStable Laws for Sums of Reciprocals
We obtain the limiting behavior of the sum of reciprocal powers of a random sample. In particular, the mean sample reciprocal tends to a Cauchy distribution centered on the principal value (PV) of the mean population reciprocal. AMS 2000 subject classification: Primary 60E07; Secondary 60F05.
متن کاملSums of Reciprocals of Polynomials over Finite Fields
Consider the following example (typical in college algebra): 1 x2 + 1 x2 + 1 + 1 x2 + x + 1 x2 + x+ 1 = 4x + 6x + 8x + 6x + 3x+ 1 x2(x+ 1) (x2 + 1) (x2 + x+ 1) . Now let’s assume that all the coefficients in the above are from the binary field F2 = Z2 = {0, 1}. The result becomes much cleaner: 1 x2 + 1 x2 + 1 + 1 x2 + x + 1 x2 + x+ 1 = 1 (x2 + x)(x4 + x) . After a brief introduction to finite f...
متن کاملOn the sum of reciprocal Tribonacci numbers
In this paper we consider infinite sums derived from the reciprocals of the Fibonacci numbers, and infinite sums derived from the reciprocals of the square of the Fibonacci numbers. Applying the floor function to the reciprocals of these sums, we obtain equalities that involve the Fibonacci numbers.
متن کاملLinear algebra and the sums of powers of integers
A general framework based on linear algebra is presented to obtain old and new polynomial expressions for the sums of powers of integers. This framework uses changes of polynomial basis, infinite lower triangular matrices and finite differences.
متن کامل