# Record Gaps between Twin Primes

## Theorem

The gaps between the following pairs of twin primes are larger than those for all smaller pairs:

 $\displaystyle \tuple {3, 5}$ $\to$ $\displaystyle \tuple {5, 7}$ a gap of $0$ $\displaystyle \tuple {5, 7}$ $\to$ $\displaystyle \tuple {11, 13}$ a gap of $4$ $\displaystyle \tuple {17, 19}$ $\to$ $\displaystyle \tuple {29, 31}$ a gap of $10$ $\displaystyle \tuple {41, 43}$ $\to$ $\displaystyle \tuple {59, 61}$ a gap of $16$ $\displaystyle \tuple {69, 73}$ $\to$ $\displaystyle \tuple {101, 103}$ a gap of $28$ $\displaystyle \tuple {311, 313}$ $\to$ $\displaystyle \tuple {347, 349}$ a gap of $34$ $\displaystyle \tuple {347, 349}$ $\to$ $\displaystyle \tuple {419, 421}$ a gap of $70$ $\displaystyle \tuple {659, 661}$ $\to$ $\displaystyle \tuple {809, 811}$ a gap of $148$

## Proof

By cases and inspection.