Record Gaps between Twin Primes

Theorem

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

\(\ds \tuple {3, 5}\)

\(\to\)

\(\ds \tuple {5, 7}\)

a gap of $0$

\(\ds \tuple {5, 7}\)

\(\to\)

\(\ds \tuple {11, 13}\)

a gap of $4$

\(\ds \tuple {17, 19}\)

\(\to\)

\(\ds \tuple {29, 31}\)

a gap of $10$

\(\ds \tuple {41, 43}\)

\(\to\)

\(\ds \tuple {59, 61}\)

a gap of $16$

\(\ds \tuple {69, 73}\)

\(\to\)

\(\ds \tuple {101, 103}\)

a gap of $28$

\(\ds \tuple {311, 313}\)

\(\to\)

\(\ds \tuple {347, 349}\)

a gap of $34$

\(\ds \tuple {347, 349}\)

\(\to\)

\(\ds \tuple {419, 421}\)

a gap of $70$

\(\ds \tuple {659, 661}\)

\(\to\)

\(\ds \tuple {809, 811}\)

a gap of $148$

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

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Proof

By cases and inspection.

Sources

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