Reflecting Numbers of Various Types, I

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 $\gcd(k,m)\ge3$.