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