Reflecting Numbers of Various Types, I

TL;DR

This paper introduces reflecting numbers, classifies them by type, and explores their properties, including their relation to congruent numbers and prime distributions, proposing conjectures and proving certain non-existence results.

Contribution

It defines reflecting numbers of various types, links them to congruent numbers, and proves new results about their distribution and properties, including infinite families and non-existence conditions.

Findings

All prime numbers p ≡ 5 mod 8 are reflecting congruent numbers.

Infinitely many square-free reflecting congruent numbers exist with specific prime divisors.

No reflecting numbers of type (k,m) exist if gcd(k,m) ≥ 3.

Abstract

The purpose of this paper is to introduce the concept of reflecting numbers to the realm of number theory and to classify reflecting numbers of certain types. For us, reflecting numbers are coming from congruent numbers, above congruent numbers, and away from congruent numbers. Explicitly speaking, a reflecting number of type $(k,m)$ is the average of two distinct rational $k$th powers, between which the distance is twice another nonzero rational $m$th power. In particular, reflecting numbers of type $(2,2)$ are all congruent numbers and thus will be called reflecting congruent numbers in this paper. We can show that all prime numbers $p\equiv5\mod8$ are reflecting congruent and in general for any integer $k\ge0$ there are infinitely many square-free reflecting congruent numbers in the residue class of $5$ modulo $8$ with exactly $k+1$ prime divisors. Moreover, we conjecture that all prime congruent numbers $p\equiv1\mod8$ are reflecting congruent. In addition, we show that there are no reflecting numbers of type $(k,m)$ if $\gcd(k,m)\ge3$.