# Record Gaps between Twin Primes

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## 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 $148$ |

This sequence is A036061 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Proof

By cases and inspection.

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $661$