Triangular Primes

Each triangular number $T_n$ is of the form $\frac{n(n+1)}{2}$ for positive integers $n$ . We see pretty quickly that $T_2 = 3$ , which is prime. Let's show that there are no larger primes in this form.

If $n$ is an odd integer greater than 2, then $\frac{n(n + 1)}{2} = \frac{(2k + 1)(2k + 2)}{2} = (2k + 1)(k + 1)$ is not prime as it is a product of two positive integers each larger than 1. A similar argument follows for even $n > 2$ .