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
Neupane et al. (Tue,) studied this question.