Lonely Twin Primes

Today I celebrate my first prime in 30 days. The previous prime was 28319 and the number associated with my diurnal age today is 28349. It's a twin prime and so 28351 is also prime. To my suprise however, the next prime after that is 36 days away. It is the number 28387 that falls on December 22nd 2026. 

Usually after a long drought of primes, there follows a flurry of them but not in this case. I was right to suspect this is an unusual occurrence. The total span of days before 28349 and after 28351 is 30 + 36 = 66. I decided to find all the record breaking twin prime pairs between 1 and one million. Here is the result where \(p\) and \(q\) are the twin primes (permalink).

  gap   p        q        gap   total 

  1     3        5        2     3

  2     5        7        4     6

  4     11       13       4     8

  6     29       31       6     12

  6     137      139      10    16

  10    419      421      10    20

  12    521      523      18    30

  22    1151     1153     10    32

  18    1931     1933     16    34

  16    1949     1951     22    38

  22    6449     6451     18    40

  34    10007    10009    28    62

  30    28349    28351    36    66

  52    35729    35731    16    68

  52    48731    48733    18    70

  22    85931    85933    58    80

  34    107507   107509   54    88

  70    173429   173431   42    112

  40    838247   838249   100   140

 If we use the smaller of the twin primes as the marker then the sequence becomes:

3, 5, 11, 29, 137, 419, 521, 1151, 1931, 1949, 6449, 10007, 28349, 35729, 48731, 85931, 107507, 173429, 838247

The sequence is not in the OEIS and I've no intention of putting it there.