Greenwood's Conjecture

Famous False Conjecture
Let $p$ be a prime number.

Let $T_p$ be the $p$th triangular number.

The conjecture states that:
 * If $T_p$ is even, then $T_p + 1$ is prime
 * If $T_p$ is odd, then $T_p - 2$ is prime.

Refutation
We have that $T_{31} = 496$, but $497 = 7 \times 71$ and so is not prime.