# 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.

## Historical Note

Greenwood's Conjecture was apparently made by a mathematician named Thomas Greenwood, but no information about him has yet been found on the internet.