
06604799
j
2016e.00610
Prakash, Anand
Mathematical curiosities: pentagonal identities.
Parabola 51, No. 1, 3 p., electronic only (2015).
2015
AMT Publishing, Australian Mathematics Trust, University of Canberra, Canberra; School of Mathematics \& Statistics, University of New South Wales, Sydney
EN
F60
I30
N70
pentagonal numbers
figurate numbers
polygonal numbers
triangular numbers
square numbers
regular polygons
Fermat's polygonal number theorem
polycat pairs
https://www.parabola.unsw.edu.au/files/articles/20102019/volume512015/issue1/vol51_no1_3.pdf
From the text: Polygonal numbers enumerate the number of points in a regular geometrical arrangement of the points in the shape of a regular polygon. An example is the triangular number $T_n$ which enumerates the number of points in a regular triangular lattice of points whose overall shape is a triangle. Square numbers $S_n$ can be defined in a similar fashion enumerating the number of points in a regular square lattice of points whose overall shape is a square. The geometrical construction for the first four pentagonal numbers is shown in a figure. Hexagonal numbers, heptagonal numbers etc. can also be defined. In general if $s$ denotes the number of sides of a regular polygon then the $n$th $s$gonal number is given by $\theta (s,n)=\frac{1}{2}(n^2(s2)n(s4))$. A curiosity considered here is that the sum of two polygonal numbers $\theta (s,n)$ and $\theta (s,m)$ is sometimes equal to the number composed of the concatenation of $n$ and $m$. For example, among the pentagonal numbers, $P_3=12$ and $P_4=22$, so that $P_3+P_4=34$, which is the concatenation of 3 and 4. We will refer to such pairs as polycat pairs.