mathschallenge.net

Problem

Given that $p$ is prime, when is 8$p$+1 a triangle number?

Solution

Let 8$p$ + 1 = $k$($k$ + 1)/2, so 16$p$ + 2 = $k$($k$ + 1)

Therefore 16$p$ = $k$² + $k$ 2 = ($k$ 1)($k$ + 2).

If $k$ 1 is odd, then $k$ + 2 will be even, and vice versa. In other words, only one of these factors is even, and so it must be a multiple of 16.

As $k$ 1 and $k$ + 2 differ by 3, we can write 16$p$ = 16$m$(16$m$ 3), leading to $p$ = $m$(16$m$ 3).

Clearly $m$ = 1, otherwise we would be dealing with a composite number, and $p$ = 13 or $p$ = 19.

When $p$ = 13, 8$p$ + 1 = 105 = $t$₁₄ and when $p$ = 19, 8$p$ + 1 = 153 = $t$₁₇.

Related problem:

Square Search: When is 8$p$+1 square?