Representation of positive integers by the form $x_1...x_k + x_1 + ... + x_k$

TL;DR

This paper investigates how positive integers can be represented using a specific algebraic form involving products and sums of ordered variables, and explores the minimal number of variables needed for such representations.

Contribution

It introduces a new problem of representing positive integers with a particular form and analyzes the minimal number of variables required for these representations.

Findings

Characterization of representations for given k

Bounds on the smallest k for representation

Conditions for the existence of representations

Abstract

For an arbitrary given $k\geq3,$ we consider a possibility of representation of a positive number $n$ by the form $x_1...x_k+x_1+...+x_k, 1\leq x_1\leq ... \leq x_k.$ We also study a question on the smallest value of $k\geq3$ in such a representation.