This analysis investigates the relationship between an odd prime p and its closest square-centered symmetric prime q, defined by q = 2n^2 - p, where n is chosen to minimize |p - q|.
The purpose of this analysis is exploratory. Linear regression is used solely as a descriptive tool to visualize the central tendency of the observed pairs, not as a proposed generative or predictive model.
- As the sample size increases, the fitted slope converges toward 1, while the intercept fluctuates around zero.
- The dispersion around the diagonal q = p increases with p, indicating non-exact but structured symmetry.
- This behavior may result from the increasing availability of admissible quadratic centers as p grows.
No claims of proof or deterministic laws are made. These results are presented to motivate further investigation.