On the Sum of two Polygonal Numbers and Conditions for Closure

Polygonal numbers arise from geometrical patterns of dots or points forming regular polygons and include known squences such as triangular and square numbers. A natural question in number theory is whether the sum of two polygonal numbers of same order itself is a polygonal number of that order. This paper studies the equationPn (a) + Pn (b) = Pn (c)where Pn(k) denotes the k-th n-gonal polygon. Using a combination of experimental observation, computational evidence, and theoretical analysis, we examine this equation for different values of n. We show that triangular and square numbers are not closed under addition , yet give infinitely many solutions, with square case precisely corresponding with pythagorean identity.By contrast , higher polygonal numbers (n ≥ 5) show rapid decline in additive structure, with solution becoming extremely rare. Computational evidence support this result and explains why the solutions dissappear as n increases