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

Searching for some patterns


A00228310^n – 1110^n + 1¬ 
A09317210^n – 3310^n + 3A159352
 ¬10^n – 7710^n + 7A159031
A093177 10^n – 9910^n + 9¬ 
 ¬10^n – 111110^n + 11¬ 
 ¬10^n – 131310^n + 13¬ 
 ¬10^n – 171710^n + 17¬ 
 ¬10^n – 191910^n + 19¬ 
 ¬10^n – 212110^n + 21A138861
 ¬10^n – 232310^n + 23¬ 
A17698710^n – 272710^n + 27¬ 
 ¬10^n – 292910^n + 29¬ 
 ¬10^n – 313110^n + 31¬ 
 ¬10^n – 333310^n + 33¬ 
 ¬10^n – 373710^n + 37¬ 
 ¬10^n – 393910^n + 39¬ 
 ¬10^n – 414110^n + 41¬ 
 ¬10^n – 434310^n + 43¬ 
 ¬10^n – 474710^n + 47¬ 
 ¬10^n – 494910^n + 49¬ 
 ¬10^n – 515110^n + 51¬ 
 ¬10^n – 535310^n + 53¬ 
A17741710^n – 575710^n + 57¬ 
A17741810^n – 595910^n + 59¬ 
 ¬10^n – 616110^n + 61¬ 
 ¬10^n – 636310^n + 63¬ 
 ¬10^n – 676710^n + 67¬ 
 ¬10^n – 696910^n + 69¬ 
 ¬10^n – 717110^n + 71¬ 
 ¬10^n – 737310^n + 73¬ 
 ¬10^n – 777710^n + 77¬ 
 ¬10^n – 797910^n + 79¬ 
 ¬10^n – 818110^n + 81¬ 
 ¬10^n – 838310^n + 83¬ 
 ¬10^n – 878710^n + 87¬ 
 ¬10^n – 898910^n + 89¬ 
 ¬10^n – 919110^n + 91¬ 
 ¬10^n – 939310^n + 93¬ 
 ¬10^n – 979710^n + 97¬ 
 ¬10^n – 999910^n + 99¬ 


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.




A000190 Prime Curios! Prime Puzzle
Wikipedia 190 Le nombre 190














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