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Polygonal numbers and sums of squares of primes are distinct fields of number theory. Here we consider sums of squares of consecutive (of order and rank) polygonal numbers. We try to express sums of squares of polygonal numbers of consecutive orders in matrix form. We also try to find the solution of a Diophantine equation in terms of polygonal numbers.

Polygonal numbers have been meticulously studied since their very beginnings in ancient Greece. Numerous discoveries stemmed from these peculiar numbers can be seen in the basic fundamental group work of number theory today with finding such as pascalâ€™s triangle and Fermat triangular number theorem. It becomes a popular field of research for mathematicians. The concept of polygonal numbers was first defined by the Greek Mathematical hypsicles in the year 170 BC. If the polygonal numbers are divided successively into triangles it will ultimately end up with right triangle. The right triangles immediately remind us of Pythagorean property. This leads to the idea of finding sums of squares of consecutive polygonal numbers. In this paper we calculate sums of squares polygonal numbers of consecutive orders. We also calculate the sums of squares of m-gonal numbers of consecutive ranks. We analyze some properties of the above.

For

are called generalized m-gonal numbers.

Also

where

Sums of Squares of Polygonal numbers of Consecutive Orders of Same Rank

Proof

Sums of squares of Polygonal numbers of Consecutive Orders in Matrix Form [

Expressing the coefficients of

The coefficient matrix

for

In general,

where

Recursive matrix form

Consider the initial matrix as the coefficients of

The elements of next order

The first two rows elements of

In general, the matrix of order

Sums of squares of Polygonal Numbers with Consecutive ranks n, n+1.

Proof

The Triple

Proof

Consider the Diophantine equation

We try for the solution in polygonal numbers.

Take

Taking

Proof

Proof

It is observed that the polygonal numbers of consecutive ranks constitute the solution of the Diophantine equation

A. Gnanam,B. Anitha, (2016) Sums of Squares of Polygonal Numbers. Advances in Pure Mathematics,06,297-301. doi: 10.4236/apm.2016.64019

16MAG_{n}: Magna Number order n.