On some interesting tasks from classical theory of numbers

TL;DR

This paper investigates specific number theory expressions involving prime powers and quadratic forms, aiming to understand their properties and classifications within classical number theory.

Contribution

It provides new insights into the properties and classifications of expressions like (p^q-1)/(p-1) and a^2+ab+b^2 involving primes and natural numbers.

Findings

Characterization of the expression (p^q-1)/(p-1) for primes p, q

Analysis of the quadratic form a^2+ab+b^2 for natural numbers a, b

Identification of special cases and properties of these expressions

Abstract

Expressions of type $(p^q-1)/(p-1)$ and $a^2+ab+b^2$, where $a, b$ are natural and $p, q$ are prime numbers, are studied.