Problems & Puzzles: Puzzles

 Puzzle 357. Consecutive reversible primes A few weeks ago, while dialoguing with a reader of this site, I came up with the following curio: 1201,  1213,  1217,  1223,  1229,  1231,  1237, are 7 consecutive primes that are also all reversible primes; the concatenations of them 1201121312171223122912311237 is a prime too, and the reverse of the concatenation   7321132192213221712131211021 is a prime too.   Questions: Find a larger case

J.K. Andersen wrote:

The first case of more than 7 primes is the 8 primes:
35547705709, 35547705727, 35547705731, 35547705749, 35547705757, 35547705827, 35547705829, 35547705841.

The reverse concatenated prime:
1485077455392850774553728507745537575077455394750774553137507745537275077455390750774553

The only other case below 10^12 is the 8 primes 184313483729 to 184313483971.

If we ignore the concatenation and its reversal, then problem 17 mentions the first case of more than 11 consecutive reversible primes below 10^12:
The 12 primes from 9387802769 to 9387803003.
This is the only case below 10^12.

***

Fred Schineder wrote:

To find the following, I did a brute force search using PARI (excluding of course numbers that began with 2,4,5,6 or 8). The odd thing is that l=7 series beginning at n=1201 is the lowest such n for ANY l>1. This series got me very curious so I did some extra searching and found one series where l=8.

Next solution for l=7 (only other solution through 70 billion)
13175207317,13175207299,13175207291,13175207237,13175207219,13175207189,13175207167

Solution for l=8 (only one through 70 billion)
35547705841,35547705829,35547705827,35547705757,35547705749,35547705731,35547705727,35547705709

Solutions for l=6: (10 under 70 billion)
36640789,36640781,36640733,36640727,36640699,36640697
196702531,196702507,196702501,196702489,196702481,196702469
1858985951,1858985941,1858985929,1858985923,1858985911,1858985899
7127289383,7127289343,7127289331,7127289307,7127289193,7127289187
7630536931,7630536923,7630536877,7630536821,7630536791,7630536737
7693557419,7693557413,7693557367,7693557353,7693557329,7693557317
11165317417,11165317393,11165317381,11165317367,11165317327,11165317313
12875565317,12875565253,12875565211,12875565193,12875565191,12875565187
13906833869,13906833859,13906833839,13906833821,13906833791,13906833769
39956136119,39956136109,39956136091,39956136079,39956136067,39956136041

Solutions for l=5: (There are 10 under 10^9)
33673903,33673897,33673883,33673879,33673877
91662583,91662581,91662553,91662551,91662509
197660431,197660417,197660383,197660381,197660357
308203877,308203873,308203867,308203859,308203853
308914427,308914391,308914379,308914373,308914367
339618589,339618583,339618581,339618533,339618527
346050641,346050637,346050599,346050583,346050577
367519027,367518997,367518973,367518929,367518923
943562443,943562387,943562371,943562369,943562329
953885423,953885411,953885363,953885353,953885347

Solutions for l=4: (There are 14 under 10^8)
376009,376003,376001,375997
1469893,1469887,1469879,1469857
12326009,12325991,12325981,12325967
12433829,12433819,12433801,12433789
17023879,17023873,17023861,17023859
19068319,19068299,19068263,19068251
31061489,31061479,31061467,31061461
34451341,34451317,34451311,34451293
74800519,74800507,74800447,74800409
77310283,77310281,77310263,77310257
95325313,95325277,95325247,95325203
96776923,96776917,96776879,96776863
98245909,98245897,98245867,98245859
98305423,98305397,98305369,98305351

Solutions for l=3: (There are 11 under 10^6)
34847,34843,34841
95803,95801,95791
137369,137363,137359
314261,314257,314243
389299,389297,389287
701401,701399,701383
928471,928469,928463
973523,973487,973459
980729,980719,980717
980957,980921,980911
983987,983951,983929

Solutions for l=2 (There are 10 under 10^5)
30529,30517
31121,31091
32491,32479
34603,34591
38651,38639
71317,71293
74077,74071
78649,78643
79769,79757
95143,95131

***

Farideh Firoozbakht wrote:

Case k=2;
1222222222444889, 1222222222444907 are two consecutive reversible primes;
the concatenation of them 12222222224448891222222222444907 and the reversal
of the concatenation 70944422222222219884442222222221 are two 32-digit primes.

Case k=4;
31061461, 31061467, 31061479, 31061489 are four consecutive reversible primes;
the concatenation of them 31061461310614673106147931061489 and the reversal
of the concatenation  98416013974160137641601316416013 are two 32-digit primes.
Case k=5;
33673877, 33673879, 33673883, 33673897, 33673903  are five  consecutive
reversible primes;  the concatenation of them
3367387733673879336738833367389733673903 and the reversal of the concatenation
3093763379837633388376339783763377837633 are two  40-digit primes.

Case k=6;
36640697, 36640699, 36640727, 36640733, 36640781, 36640789, 79604663 are six
consecutive reversible primes; the concatenation of them
366406973664069936640727366407333664078136640789 and the reversal of the
concatenation 987046631870466333704663727046639960466379604663 are two
48-digit  primes.

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