When is the difference between two triangular numbers a prime number? When is the difference between two triangular numbers a prime number? and what is the rule? I have tried drawing it out,graphs and tables however I have been unsuccessful in finding an answer.
Source: AS Level maths

You would do better to note that a triangle number is of the form $\cfrac {n(n-1)}2$ and explore the difference of two triangle numbers with $n\gt m$ which is $$\cfrac {n(n-1)}2-\cfrac {m(m-1)}2$$ Hint: can you factorise the expression which arises? You will need to take care with the factor $2$ in the denominator and deal with the case of the prime being $2$, which is even, but small enough to look at on its own.
