Each of the numbers
$$
1=1,3=1+2,6=1+2+3,10=1+2+3+4, \ldots
$$
represents the number of dots that can be arranged evenly in an equilateral triangle:
$$
\therefore \cdots
$$
This led the ancient Greeks to call a number triangular if it is the sum of consecutive integers, beginning with 1 . Prove the following facts concerning triangular numbers:
(a) A number is triangular if and only if it is of the form $n(n+1) / 2$ for some $n \geq 1$. (Pythagoras, circa 550 B.C. $)$
(b) The integer $n$ is a triangular number if and only if $8 n+1$ is a perfect square. (Plutarch, circa 100 A.D.)
(c) The sum of any two consecutive triangular numbers is a perfect square. (Nicomachus, circa $100 \mathrm{A.D}$.)
(d) If $n$ is a triangular number, then so are $9 n+1,25 n+3$, and $49 n+6 .$ (Euler, 1775 )
Divisibility Theory in the Integers
Early Number Theory