Chapter 28

Polygonal triples

We consider polygonal numbers of a fixed shape. For a given positive
integerk, the sequence ofk-gonal numbers consists of the integers

Pk,n:=

1

2

(

(k−2)n^2 −(k−4)n

)

. (28.1)

By ak-gonal triple, we mean a triple of positive integers(a, b, c)satis-
fying
Pk,a+Pk,b=Pk,c. (28.2)

A 4 -gonal triple is simply a Pythagorean triple satisfyinga^2 +b^2 =c^2.
We shall assume thatk=4. By completing squares, we rewrite (28.2)
as

(2(k−2)a−(k−4))^2 +(2(k−2)b−(k−4))^2

=(2(k−2)c−(k−4))^2 +(k−4)^2 , (28.3)

and note, by dividing throughout by(k−4)^2 , that this determines a
rationalpoint on the surfaceS:

x^2 +y^2 =z^2 +1, (28.4)

namely,
P(k;a, b, c):=(ga− 1 ,gb− 1 ,gc−1), (28.5)

whereg=2(kk−− 4 2). This is always an integer point fork=3, 5 , 6 , 8 , with
correspondingg=− 2 , 6 , 4 , 3 .Fork=3(triangular numbers), we shall
change signs, and consider instead the point

P′(3;a, b, c):=(2a+1, 2 b+1, 2 c+1). (28.6)

The coordinates ofP′(3;a, b, c)are all odd integers exceeding 1.