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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/2010-2019/volume-51-2015/issue-1/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 $\theta (s,n)=\frac{1}{2}(n^2(s-2)-n(s-4))$. 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.