Sign in to use this feature.

Years

Between: -

Subjects

remove_circle_outline
remove_circle_outline
remove_circle_outline
remove_circle_outline

Journals

Article Types

Countries / Regions

Search Results (2)

Search Parameters:
Keywords = Nicomachus theorem

Order results
Result details
Results per page
Select all
Export citation of selected articles as:
15 pages, 673 KB  
Article
Analytical Relations and Statistical Estimations for Sums of Powered Integers
by Stan Lipovetsky
Axioms 2025, 14(1), 30; https://doi.org/10.3390/axioms14010030 - 1 Jan 2025
Viewed by 1390
Abstract
Finding analytical closed-form solutions for the sums of powers of the first n positive integers is a classical problem of number theory. Analytical methods of constructing such sums produce complicated formulae of polynomials of a higher order, but they can be presented via [...] Read more.
Finding analytical closed-form solutions for the sums of powers of the first n positive integers is a classical problem of number theory. Analytical methods of constructing such sums produce complicated formulae of polynomials of a higher order, but they can be presented via the first two power sums. The current paper describes new presentations of the power sums and their extensions from polynomial to algebraic functions. Particularly, it shows that power sums of any higher order can be expressed just by a value of the arithmetic progression of the first power sum, or by the second power sum, or approximately by any another power sum. Regression modeling for the estimation of the powered sums is also considered, which is helpful for finding approximate values of long sums for big powers. Several problems based on the relations between sums of different powers in explicit forms are suggested for educational purposes. Full article
Show Figures

Figure 1

31 pages, 22965 KB  
Article
Some Elementary Aspects of 4-Dimensional Geometry
by J. Scott Carter and David A. Mullens
Symmetry 2015, 7(2), 515-545; https://doi.org/10.3390/sym7020515 - 4 May 2015
Viewed by 8836
Abstract
We indicate that Heron’s formula (which relates the square of the area of a triangle to a quartic function of its edge lengths) can be interpreted as a scissors congruence in four-dimensional space. In the process of demonstrating this, we examine a number [...] Read more.
We indicate that Heron’s formula (which relates the square of the area of a triangle to a quartic function of its edge lengths) can be interpreted as a scissors congruence in four-dimensional space. In the process of demonstrating this, we examine a number of decompositions of hypercubes, hyper-parallelograms and other elementary four-dimensional solids. Full article
(This article belongs to the Special Issue Diagrams, Topology, Categories and Logic)
Show Figures

Back to TopTop