Additive and Multiplicative Operations on Set of Polygonal Numbers

Abstract

In this paper, the focus on generating k-gonal numbers P (k, n) \(=\) \(\left\{ \begin{array}{r} \frac{\mathbf{n}}{\mathbf{2}}\left\lbrack \left( \mathbf{k - 3} \right)\left( \mathbf{n - 1} \right)\mathbf{+}\left( \mathbf{n + 1} \right) \right\rbrack\mathbf{\ for\ k > 2,n \geq 0} \\ \frac{\mathbf{n}}{\mathbf{2}}\left\lbrack \left( \mathbf{k - 3} \right)\left( \mathbf{n + 1} \right)\mathbf{+}\left( \mathbf{n - 1} \right) \right\rbrack\mathbf{\ for\ k > 2,n < 0} \end{array} \right\}\). Also introduce to define additive (+) and multiplicative (*) operations on Sets of k-gonal numbers. In particular, concerning addition (+), \(\mathbf{p}\left( \mathbf{k,m} \right)\mathbf{+ p}\left( \mathbf{k,n} \right)\mathbf{= p}\left( \mathbf{k,m + n} \right)\mathbf{-}\left( \mathbf{k - 2} \right)\mathbf{mn\ \ }forsome\ integer\mathbf{\ k > 2}\). Also, concerning multiplication (*), \(\mathbf{p}\left( \mathbf{k,m} \right)\mathbf{*p}\left( \mathbf{k,n} \right)\mathbf{= p}\left( \mathbf{k,mn} \right)\mathbf{+}\frac{\left( \mathbf{k - 4} \right)\left( \mathbf{k - 2} \right)\mathbf{mn}}{\mathbf{4}}\left( \mathbf{m - 1} \right)\left( \mathbf{n - 1} \right)\mathbf{\ }forsome\ integer\mathbf{\ \ k > 2}\).Also, summation of any two different k-gonal is \(\mathbf{p}\left( \mathbf{k}_{\mathbf{1}}\mathbf{,n} \right)\mathbf{+ p}\left( \mathbf{k}_{\mathbf{2}}\mathbf{,n} \right)\mathbf{= n}\left( \mathbf{n + 1} \right)\mathbf{+}\frac{\mathbf{n}\left( \mathbf{n - 1} \right)}{\mathbf{2}}\left\lbrack \left( \mathbf{k}_{\mathbf{1}}\mathbf{+}\mathbf{k}_{\mathbf{2}} \right)\mathbf{- 6} \right\rbrack\mathbf{\ for\ }\mathbf{k}_{\mathbf{1}}\mathbf{,}\mathbf{k}_{\mathbf{2}}\mathbf{> 2}\). Also, I applied above properties on some Sets of Polygonal numbers, which are generated by replacing integer k with 3,4,5,6,7 and 8. Also, introduced to study of Repeated steps of Residues of the above Sets of Polygonal numbers generated by k with 3,4,5,6,7 and 8.