[ November 29, 2014 ]
OEIS Reference Table for primes of the form 10^n –/+ d
Searching for some patterns

Detecting patterns

The midcolumn reveals immediately some patterns.
When there exist more than one prime for 10^m – d
and more than one prime for 10^n + d then the displacements d
are all congruent to 3 mod 6 (or to 0 mod 3).
The sequence (highlighted with cells in light green background color) looks like this :
3, 9, 21, 27, 33, 39, 51, 57, 63, 69, 81, 87, 93, 99
{ After the initial 3 only odd composites divisible by 3 appear. }
A second pattern pops up when jumping from one to the next term :
+6, +12, +6, +6, +6, +12, +6, +6, +6, +12, +6, +6, +6
Is this [+6, +12, +6, +6] an infinite pattern ?

When there exist more than one prime for 10^m – d
but not for 10^n + d then the displacements d
are all congruent to 5 mod 6.
The sequence (aligned leftwards in the midcolumn of the table) looks like this :
11, 17, 23, 29, 41, 47, 53, 59, 71, 77, 83, 89
{ only one composite in the list... }
An analogue second pattern shows up :
+6, +6, +6, +12, +6, +6, +6, +12, +6, +6, +6
Same question : is this [+6, +6, +6, +12] an infinite pattern ?

When there exist more than one prime for 10^n + d
but not for 10^m – d then the displacements d
are all congruent to 1 mod 6.
The sequence (aligned rightwards in the midcolumn of the table) looks like this :
1, 7, 13, 19, 31, 37, 43, 49, 61, 67, 73, 79, 91, 97
{ Equivalent to A004611 - Divisible only by primes congruent to 1 mod 3. }
An analogue second pattern shows up :
+6, +6, +6, +12, +6, +6, +6, +12, +6, +6, +6, +12, +6
Same question : is this [+6, +6, +6, +12] an infinite pattern ?

ps. displacements that are multiples of 5 are not considered.
These can never give rise to more than one prime, namely 5 itself, in either forms.

If there is more to say about this topic please write me and I'll add your comments.