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Problem

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

Solution

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

Therefore 16p = k² + 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 16p = 16m(16m 3), leading to p = m(16m 3).

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

When p = 13, 8p + 1 = 105 = t₁₄ and when p = 19, 8p + 1 = 153 = t₁₇.

Related problem:

Square Search: When is 8p+1 square?