Bounded Relative Distances of Symmetric Prime Pairs Around Even Integers

This paper investigates the distribution of symmetric prime pairs around the midpoint of even integers. For a given even integer $2n$, prime pairs $(p_1,p_2)$ symmetrically positioned about $n$ are considered, satisfying

$$
n-p_1 = p_2-n,
$$

or equivalently,

$$
p_1+p_2=2n.
$$

Particular attention is given to the nearest symmetric prime pair associated with each even integer. Using the symmetric distance

$$
d = |n-p_i|,\qquad i=1,2,
$$

the relative distance ratio

$$
R_n=\frac{d}{n}
$$

is introduced as a normalized measure of how far the nearest symmetric prime pair lies from the midpoint. Computational experiments were carried out for selected even integers up to $10^6$. The resulting data exhibit bounded behavior within the tested range and suggest an empirical asymptotic decay of $R_n$ toward zero as $n$ increases, except in cases where the midpoint itself is prime, for which $R_n=0$. Within the computed data, the largest observed ratio occurs at $2n=44$, where

$$
R_{22}=\frac{9}{22}=0.4090909\ldots.
$$

The paper presents a ratio-based framework for studying local symmetric prime distributions and highlights open questions concerning upper bounds, asymptotic estimates, and the behavior of additional symmetric prime pairs located farther from the midpoint.