Problems & Puzzles: Puzzles

Puzzle 1216 Related to Puzzle 1212 In this puzzle I will ask to insert a well defined string in-between each pair of consecutve digits of an integer p, according to the rule described in the sequence A381732, (Puzzle 1212). After making all the possible insertions in p, you will have a new integer q. Example: p=1693 (prime) q= 123456789876543 (composite) Here are the first nine p values such that p & q are both prime integers: p q 29 23456789 47 4567 73 76543 113 1123 239 23456789 269 23456789 353 34543 373 345676543 383 34567876543 In the table above I have used bold & blue characters only for these pairs (p, q) where both are palprimes Q1. Find the first 20 pais (p, q) where both are palprimes Q2. Send the largest q you can get where both p & q are palprimes.

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During the week 11-17 Abril 2025, contributions came from Gennady Gusev, Paul Cleary, Michael Branicky, Emmanuel Vantieghem, Jeff Heleen, Giorgos Kalogeropoulos, Simon Cavegn

 

***

Gennady wrote:

Q1.

The First 3-, 5- and 7-digit palprimes of this type:

353, 34543

373, 345676543

383, 34567876543

11311, 1123211

16561, 1234565654321

30803, 32101234567876543210123

34843, 34567876543

35053, 345432101234543

35753, 345676543

36563, 345656543

38083, 345678765432101234567876543

38183, 3456787654321234567876543

38783, 3456787876543

1317131, 123212345676543212321

1328231, 1232345678765432321

1565651, 1234565654321

1597951, 123456789878987654321

1987891, 123456789878987654321

3064603, 32101234565456543210123

3252523, 323454323454323

3331333, 333212333

3470743, 34567654321012345676543

3487843, 3456787876543

3515153, 345432123454321234543

3601063, 345654321010123456543

3670763, 34567654321012345676543

3680863, 345678765432101234567876543

3698963, 345678989876543

3709073, 34567654321012345678987654321012345676543

3781873, 3456787654321234567876543

3871783, 3456787654321234567876543

3997993, 3456789987899876543

7093907, 765432101234567898765434567898765432101234567

7576757, 76567676567

7662667, 7665432345667

7938397, 789876543456787654345678987

7949497, 7898765456789876545678987

9127219, 98765432123456765432123456789

9212129, 9876543212123456789

9217129, 98765432123456765432123456789

9817189, 98765432123456765432123456789

9927299, 998765432345676543234567899

 

Q2.

p=321933151336330776721924306926918730333921317363165358903903740375996168321738711135352329930357144916713940
96615515617194098430215732477418333812010933837372212418236411114439919410538773531611110130916636017031012536
63251401241491481131033331111011861991021081051011452541101501801201991681101111333301311841941421041523663521
01307106366190310111161353778350149199344111146328142122737383390102183338147742375120348904917165155166904931
7619441753039923253531117837123861699573047309309853561363713129333037819629603429127677033633151339123 (541 digits),
q=32123456789876543321234543212334565433210123456776765432123456789876543234321012345678987654323456789876543
2123456787654321012333456789876543212321234567654345654321234565434567898765432101234567898765432101234567654
3210123456765678998765432123456787654321234567654345678765432111234543454323234567899876543210123456765432123
4456789876543212345676543212345678987654321012345678987665432123455432123456543212345676543212345678987654321
0123456789876543210121234567654323456776543212345678765433345678765432121010123456789876543345678765434567654
3456765432212343212345678765432345654321111234434567899876543212345678987654321012345434567877654345432123456
5432111101232101234567898765432123456654345654321012345676543210123210123454345665432345432123432101234321234
5678987654321234567876543211232101233332111101123456787654321234567899876543210121012345678765432101234543210
1123454323454321101234543210123456787654321012101234567899876543212345678765432110111123333210123211234567876
5432123456789876543212343210123432123454323456654345432101232101234567654321012345654345665432123456789876543
2101232101111234565432123454345677876543454321012345678987654321234567899876543443211112345654323456787654321
2343212234567654345676543456787654334567898765432101012123456787654333456787654321234567765432345676543212101
2345678987654321012345678987654321234567654321234565432123455432123456678987654321012345678987654321234567654
3212345678987654432123456765432101234567899876543232345434543211123456787654345676543212345678765432123456789
9876567654321012345676543210123456789876543210123456789876543456543212345654345676543212321234567898765433321
0123456787654321234567898765432345678987654321012343234567898765432123456767765432101233456543321234543212334
567898765432123 (1757 digits).

(Palprime p is from my solution of the puzzle 1143).

***

Paul wrote:

Q1.

353  Palindromic Prime: 34543 (Digit Length: 5)
373  Palindromic Prime: 345676543 (Digit Length: 9)
383  Palindromic Prime: 34567876543 (Digit Length: 11)
11311  Palindromic Prime: 1123211 (Digit Length: 7)
16561  Palindromic Prime: 1234565654321 (Digit Length: 13)
30803  Palindromic Prime: 32101234567876543210123 (Digit Length: 23)
34843  Palindromic Prime: 34567876543 (Digit Length: 11)
35053  Palindromic Prime: 345432101234543 (Digit Length: 15)
35753  Palindromic Prime: 345676543 (Digit Length: 9)
36563  Palindromic Prime: 345656543 (Digit Length: 9)
38083  Palindromic Prime: 345678765432101234567876543 (Digit Length: 27)
38183  Palindromic Prime: 3456787654321234567876543 (Digit Length: 25)
38783  Palindromic Prime: 3456787876543 (Digit Length: 13)
1317131  Palindromic Prime: 123212345676543212321 (Digit Length: 21)
1328231  Palindromic Prime: 1232345678765432321 (Digit Length: 19)
1565651  Palindromic Prime: 1234565654321 (Digit Length: 13)
1597951  Palindromic Prime: 123456789878987654321 (Digit Length: 21)
1987891  Palindromic Prime: 123456789878987654321 (Digit Length: 21)
3064603  Palindromic Prime: 32101234565456543210123 (Digit Length: 23)
3252523  Palindromic Prime: 323454323454323 (Digit Length: 15)
3331333  Palindromic Prime: 333212333 (Digit Length: 9)
3470743  Palindromic Prime: 34567654321012345676543 (Digit Length: 23)
3487843  Palindromic Prime: 3456787876543 (Digit Length: 13)
3515153  Palindromic Prime: 345432123454321234543 (Digit Length: 21)
3601063  Palindromic Prime: 345654321010123456543 (Digit Length: 21)
3670763  Palindromic Prime: 34567654321012345676543 (Digit Length: 23)
3680863  Palindromic Prime: 345678765432101234567876543 (Digit Length: 27)
3698963  Palindromic Prime: 345678989876543 (Digit Length: 15)
3709073  Palindromic Prime: 34567654321012345678987654321012345676543 (Digit Length: 41)
3781873  Palindromic Prime: 3456787654321234567876543 (Digit Length: 25)
3871783  Palindromic Prime: 3456787654321234567876543 (Digit Length: 25)
3997993  Palindromic Prime: 3456789987899876543 (Digit Length: 19)
7093907  Palindromic Prime: 765432101234567898765434567898765432101234567 (Digit Length: 45)
7576757  Palindromic Prime: 76567676567 (Digit Length: 11)
7662667  Palindromic Prime: 7665432345667 (Digit Length: 13)
7938397  Palindromic Prime: 789876543456787654345678987 (Digit Length: 27)
7949497  Palindromic Prime: 7898765456789876545678987 (Digit Length: 25)
9127219  Palindromic Prime: 98765432123456765432123456789 (Digit Length: 29)
9212129  Palindromic Prime: 9876543212123456789 (Digit Length: 19)
9217129  Palindromic Prime: 98765432123456765432123456789 (Digit Length: 29)
9817189  Palindromic Prime: 98765432123456765432123456789 (Digit Length: 29)
9927299  Palindromic Prime: 998765432345676543234567899 (Digit Length: 27)

Q2.

Here are some with a digit length >100.  The biggest having 117 digits.

10908082928080901  Palindromic Prime: 10123456789876543210123456787654321012345678765432345678987654323456787654321012345678765432101234567898765432101 (Digit Length: 113)
11709061916090711  Palindromic Prime: 1123456765432101234567898765432101234565432123456789876543212345654321012345678987654321012345676543211 (Digit Length: 103)
11739190909193711  Palindromic Prime: 1123456765434567898765432123456789876543210123456789876543210123456789876543212345678987654345676543211 (Digit Length: 103)
11818071817081811  Palindromic Prime: 11234567876543212345678765432101234567654321234567876543212345676543210123456787654321234567876543211 (Digit Length: 101)
12609190609190621  Palindromic Prime: 1234565432101234567898765432123456789876543210123456543210123456789876543212345678987654321012345654321 (Digit Length: 103)
12808193939180821  Palindromic Prime: 12345678765432101234567876543212345678987654345678987654345678987654321234567876543210123456787654321 (Digit Length: 101)
12901809190810921  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
12908096169080921  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
13590809190809531  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
13903809190830931  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
13906809190860931  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
13908029192080931  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
13908092129080931  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
13908095159080931  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
13908509190580931  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
14083809190838041  Palindromic Prime: 12343210123456787654345678765432101234567898765432123456789876543210123456787654345678765432101234321 (Digit Length: 101)
14191919391919141  Palindromic Prime: 123432123456789876543212345678987654321234567898765434567898765432123456789876543212345678987654321234321 (Digit Length: 105)
14808061916080841  Palindromic Prime: 12345678765432101234567876543210123456543212345678987654321234565432101234567876543210123456787654321 (Digit Length: 101)
14908094149080941  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
14919072727091941  Palindromic Prime: 12345678987654321234567898765432101234567654323456765432345676543210123456789876543212345678987654321 (Digit Length: 101)
14919191719191941  Palindromic Prime: 1234567898765432123456789876543212345678987654321234567654321234567898765432123456789876543212345678987654321 (Digit Length: 109)
14939180708193941  Palindromic Prime: 12345678987654345678987654321234567876543210123456765432101234567876543212345678987654345678987654321 (Digit Length: 101)
15061809190816051  Palindromic Prime: 123454321012345654321234567876543210123456789876543212345678987654321012345678765432123456543210123454321 (Digit Length: 105)
15190609590609151  Palindromic Prime: 12345432123456789876543210123456543210123456789876567898765432101234565432101234567898765432123454321 (Digit Length: 101)
15190709790709151  Palindromic Prime: 12345432123456789876543210123456765432101234567898789876543210123456765432101234567898765432123454321 (Digit Length: 101)
15907809190870951  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
15908709190780951  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
15910809190801951  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
15917290909271951  Palindromic Prime: 123456789876543212345676543234567898765432101234567898765432101234567898765432345676543212345678987654321 (Digit Length: 105)
16060509190506061  Palindromic Prime: 12345654321012345654321012345432101234567898765432123456789876543210123454321012345654321012345654321 (Digit Length: 101)
16069190609196061  Palindromic Prime: 1234565432101234567898765432123456789876543210123456543210123456789876543212345678987654321012345654321 (Digit Length: 103)
16091902620919061  Palindromic Prime: 1234565432101234567898765432123456789876543210123456543210123456789876543212345678987654321012345654321 (Digit Length: 103)
16091905650919061  Palindromic Prime: 1234565432101234567898765432123456789876543210123456543210123456789876543212345678987654321012345654321 (Digit Length: 103)
16091930603919061  Palindromic Prime: 1234565432101234567898765432123456789876543210123456543210123456789876543212345678987654321012345654321 (Digit Length: 103)
16890809190809861  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
16908093139080961  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
16919072727091961  Palindromic Prime: 12345678987654321234567898765432101234567654323456765432345676543210123456789876543212345678987654321 (Digit Length: 101)
16960809190806961  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
17050807270805071  Palindromic Prime: 1234567654321012345432101234567876543210123456765432345676543210123456787654321012345432101234567654321 (Digit Length: 103)
17060938083906071  Palindromic Prime: 12345676543210123456543210123456789876543456787654321012345678765434567898765432101234565432101234567654321 (Digit Length: 107)
17070800700807071  Palindromic Prime: 1234567654321012345676543210123456787654321001234567654321001234567876543210123456765432101234567654321 (Digit Length: 103)
17090605350609071  Palindromic Prime: 12345676543210123456789876543210123456543210123454345432101234565432101234567898765432101234567654321 (Digit Length: 101)
17091922622919071  Palindromic Prime: 12345676543210123456789876543212345678987654322345654322345678987654321234567898765432101234567654321 (Digit Length: 101)
17181928582918171  Palindromic Prime: 12345676543212345678765432123456789876543234567876567876543234567898765432123456787654321234567654321 (Digit Length: 101)
17191949394919171  Palindromic Prime: 123456765432123456789876543212345678987654567898765434567898765456789876543212345678987654321234567654321 (Digit Length: 105)
17196908180969171  Palindromic Prime: 12345676543212345678987678987654321012345678765432123456787654321012345678987678987654321234567654321 (Digit Length: 101)
17288090709088271  Palindromic Prime: 1234567654323456788765432101234567898765432101234567654321012345678987654321012345678876543234567654321 (Digit Length: 103)
17370908180907371  Palindromic Prime: 1234567654345676543210123456789876543210123456787654321234567876543210123456789876543210123456765434567654321 (Digit Length: 109)
17391608180619371  Palindromic Prime: 12345676543456789876543212345654321012345678765432123456787654321012345654321234567898765434567654321 (Digit Length: 101)
17806091719060871  Palindromic Prime: 12345678765432101234565432101234567898765432123456765432123456789876543210123456543210123456787654321 (Digit Length: 101)
17808061916080871  Palindromic Prime: 12345678765432101234567876543210123456543212345678987654321234565432101234567876543210123456787654321 (Digit Length: 101)
17906080608060971  Palindromic Prime: 1234567898765432101234565432101234567876543210123456543210123456787654321012345654321012345678987654321 (Digit Length: 103)
17907809190870971  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
17908029192080971  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
17908098189080971  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
17908709190780971  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
17970809190807971  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
18019290409291081  Palindromic Prime: 1234567876543210123456789876543234567898765432101234321012345678987654323456789876543210123456787654321 (Digit Length: 103)
18049290809294081  Palindromic Prime: 123456787654321012345678987654323456789876543210123456787654321012345678987654323456789876543210123456787654321 (Digit Length: 111)
18060971717906081  Palindromic Prime: 12345678765432101234565432101234567898765432123456765432123456789876543210123456543210123456787654321 (Digit Length: 101)
18074939193947081  Palindromic Prime: 12345678765432101234567654567898765434567898765432123456789876543456789876545676543210123456787654321 (Digit Length: 101)
18078061916087081  Palindromic Prime: 12345678765432101234567876543210123456543212345678987654321234565432101234567876543210123456787654321 (Digit Length: 101)
18080621912608081  Palindromic Prime: 12345678765432101234567876543210123456543212345678987654321234565432101234567876543210123456787654321 (Digit Length: 101)
18080708680708081  Palindromic Prime: 123456787654321012345678765432101234567654321012345678767876543210123456765432101234567876543210123456787654321 (Digit Length: 111)
18081793939718081  Palindromic Prime: 12345678765432101234567876543212345678987654345678987654345678987654321234567876543210123456787654321 (Digit Length: 101)
18083729092738081  Palindromic Prime: 12345678765432101234567876543456765432345678987654321012345678987654323456765434567876543210123456787654321 (Digit Length: 107)
18083809590838081  Palindromic Prime: 1234567876543210123456787654345678765432101234567898765678987654321012345678765434567876543210123456787654321 (Digit Length: 109)
18083907270938081  Palindromic Prime: 123456787654321012345678765434567898765432101234567654323456765432101234567898765434567876543210123456787654321 (Digit Length: 111)
18085193939158081  Palindromic Prime: 12345678765432101234567876543212345678987654345678987654345678987654321234567876543210123456787654321 (Digit Length: 101)
18090818981809081  Palindromic Prime: 12345678765432101234567898765432101234567876543212345678987654321234567876543210123456789876543210123456787654321 (Digit Length: 113)
18091506160519081  Palindromic Prime: 123456787654321012345678987654321234543210123456543212345654321012345432123456789876543210123456787654321 (Digit Length: 105)
18091881718819081  Palindromic Prime: 12345678765432101234567898765432123456788765432123456765432123456788765432123456789876543210123456787654321 (Digit Length: 107)
18092081918029081  Palindromic Prime: 12345678765432101234567898765432101234567876543212345678987654321234567876543210123456789876543210123456787654321 (Digit Length: 113)
18092890809829081  Palindromic Prime: 123456787654321012345678987654323456789876543210123456787654321012345678987654323456789876543210123456787654321 (Digit Length: 111)
18092930403929081  Palindromic Prime: 1234567876543210123456789876543234567898765432101234321012345678987654323456789876543210123456787654321 (Digit Length: 103)
18095709290759081  Palindromic Prime: 12345678765432101234567898765676543210123456789876543234567898765432101234567656789876543210123456787654321 (Digit Length: 107)
18095809690859081  Palindromic Prime: 1234567876543210123456789876567876543210123456789876789876543210123456787656789876543210123456787654321 (Digit Length: 103)
18097081918079081  Palindromic Prime: 12345678765432101234567898765432101234567876543212345678987654321234567876543210123456789876543210123456787654321 (Digit Length: 113)
18109081918090181  Palindromic Prime: 12345678765432101234567898765432101234567876543212345678987654321234567876543210123456789876543210123456787654321 (Digit Length: 113)
18116090909061181  Palindromic Prime: 12345678765432112345654321012345678987654321012345678987654321012345678987654321012345654321123456787654321 (Digit Length: 107)
18141909590914181  Palindromic Prime: 12345678765432123432123456789876543210123456789876567898765432101234567898765432123432123456787654321 (Digit Length: 101)
18170391719307181  Palindromic Prime: 12345678765432123456765432101234567898765432123456765432123456789876543210123456765432123456787654321 (Digit Length: 101)
18170891719807181  Palindromic Prime: 12345678765432123456765432101234567898765432123456765432123456789876543210123456765432123456787654321 (Digit Length: 101)
18189380908398181  Palindromic Prime: 12345678765432123456789876543456787654321012345678987654321012345678765434567898765432123456787654321 (Digit Length: 101)
18190619991609181  Palindromic Prime: 1234567876543212345678987654321012345654321234567899987654321234565432101234567898765432123456787654321 (Digit Length: 103)
18192990709929181  Palindromic Prime: 12345678765432123456789876543234567899876543210123456765432101234567899876543234567898765432123456787654321 (Digit Length: 107)
18193905250939181  Palindromic Prime: 1234567876543212345678987654345678987654321012345432345432101234567898765434567898765432123456787654321 (Digit Length: 103)
18209290809290281  Palindromic Prime: 123456787654321012345678987654323456789876543210123456787654321012345678987654323456789876543210123456787654321 (Digit Length: 111)
18217091719071281  Palindromic Prime: 12345678765432123456765432101234567898765432123456765432123456789876543210123456765432123456787654321 (Digit Length: 101)
18240809390804281  Palindromic Prime: 12345678765432343210123456787654321012345678987654345678987654321012345678765432101234323456787654321 (Digit Length: 101)
18280480908408281  Palindromic Prime: 123456787654323456787654321012345678765432101234567898765432101234567876543210123456787654323456787654321 (Digit Length: 105)
18280804940808281  Palindromic Prime: 123456787654323456787654321012345678765432101234567898765432101234567876543210123456787654323456787654321 (Digit Length: 105)
18280870907808281  Palindromic Prime: 123456787654323456787654321012345678765432101234567898765432101234567876543210123456787654323456787654321 (Digit Length: 105)
18282080908028281  Palindromic Prime: 123456787654323456787654321012345678765432101234567898765432101234567876543210123456787654323456787654321 (Digit Length: 105)
18293939193939281  Palindromic Prime: 123456787654323456789876543456789876543456789876543212345678987654345678987654345678987654323456787654321 (Digit Length: 105)
18380718181708381  Palindromic Prime: 123456787654345678765432101234567654321234567876543212345678765432123456765432101234567876543456787654321 (Digit Length: 105)
18390837173809381  Palindromic Prime: 12345678765434567898765432101234567876543456765432123456765434567876543210123456789876543456787654321 (Digit Length: 101)
18408193939180481  Palindromic Prime: 12345678765432101234567876543212345678987654345678987654345678987654321234567876543210123456787654321 (Digit Length: 101)
18409081918090481  Palindromic Prime: 12345678765432101234567898765432101234567876543212345678987654321234567876543210123456789876543210123456787654321 (Digit Length: 113)
18419380908391481  Palindromic Prime: 12345678765432123456789876543456787654321012345678987654321012345678765434567898765432123456787654321 (Digit Length: 101)
18709290409290781  Palindromic Prime: 1234567876543210123456789876543234567898765432101234321012345678987654323456789876543210123456787654321 (Digit Length: 103)
18808092429080881  Palindromic Prime: 12345678876543210123456787654321012345678987654323432345678987654321012345678765432101234567887654321 (Digit Length: 101)
18906080608060981  Palindromic Prime: 1234567898765432101234565432101234567876543210123456543210123456787654321012345654321012345678987654321 (Digit Length: 103)
18908096169080981  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
18917290909271981  Palindromic Prime: 123456789876543212345676543234567898765432101234567898765432101234567898765432345676543212345678987654321 (Digit Length: 105)
18939180708193981  Palindromic Prime: 12345678987654345678987654321234567876543210123456765432101234567876543212345678987654345678987654321 (Digit Length: 101)
19006809190860091  Palindromic Prime: 1234567898765432100123456787654321012345678987654321234567898765432101234567876543210012345678987654321 (Digit Length: 103)
19008029192080091  Palindromic Prime: 1234567898765432100123456787654321012345678987654321234567898765432101234567876543210012345678987654321 (Digit Length: 103)
19008095159080091  Palindromic Prime: 1234567898765432100123456787654321012345678987654321234567898765432101234567876543210012345678987654321 (Digit Length: 103)
19008309190380091  Palindromic Prime: 1234567898765432100123456787654321012345678987654321234567898765432101234567876543210012345678987654321 (Digit Length: 103)
19018095159081091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19027809190872091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19029091619092091  Palindromic Prime: 1234567898765432101234567898765432101234567898765432123456543212345678987654321012345678987654321012345678987654321 (Digit Length: 115)
19038029192083091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19038092129083091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19047809190874091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19056190509165091  Palindromic Prime: 12345678987654321012345654321234567898765432101234543210123456789876543212345654321012345678987654321 (Digit Length: 101)
19058039193085091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19060070907006091  Palindromic Prime: 12345678987654321012345654321001234567654321012345678987654321012345676543210012345654321012345678987654321 (Digit Length: 107)
19060810601806091  Palindromic Prime: 1234567898765432101234565432101234567876543210123456543210123456787654321012345654321012345678987654321 (Digit Length: 103)
19061972927916091  Palindromic Prime: 12345678987654321012345654321234567898765432345678987654323456789876543212345654321012345678987654321 (Digit Length: 101)
19068509190586091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19070609690607091  Palindromic Prime: 123456789876543210123456765432101234565432101234567898767898765432101234565432101234567654321012345678987654321 (Digit Length: 111)
19070711711707091  Palindromic Prime: 1234567898765432101234567654321012345676543211234567654321123456765432101234567654321012345678987654321 (Digit Length: 103)
19070846064807091  Palindromic Prime: 1234567898765432101234567654321012345678765456543210123456545678765432101234567654321012345678987654321 (Digit Length: 103)
19070949194907091  Palindromic Prime: 123456789876543210123456765432101234567898765456789876543212345678987654567898765432101234567654321012345678987654321 (Digit Length: 117)
19073906360937091  Palindromic Prime: 123456789876543210123456765434567898765432101234565434565432101234567898765434567654321012345678987654321 (Digit Length: 105)
19078079197087091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19078409190487091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19080359195308091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19080479197408091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19080593639508091  Palindromic Prime: 1234567898765432101234567876543210123456789876543456543456789876543210123456787654321012345678987654321 (Digit Length: 103)
19080692129608091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19080696169608091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19080965156908091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19083927572938091  Palindromic Prime: 12345678987654321012345678765434567898765432345676567654323456789876543456787654321012345678987654321 (Digit Length: 101)
19084098189048091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19084209190248091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19085309190358091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19086098189068091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19087109190178091  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19088181718188091  Palindromic Prime: 12345678987654321012345678876543212345678765432123456765432123456787654321234567887654321012345678987654321 (Digit Length: 107)
19090856165809091  Palindromic Prime: 123456789876543210123456789876543210123456787656543212345656787654321012345678987654321012345678987654321 (Digit Length: 105)
19090951615909091  Palindromic Prime: 1234567898765432101234567898765432101234567898765432123456543212345678987654321012345678987654321012345678987654321 (Digit Length: 115)
19091191319119091  Palindromic Prime: 12345678987654321012345678987654321123456789876543212321234567898765432112345678987654321012345678987654321 (Digit Length: 107)
19091450705419091  Palindromic Prime: 12345678987654321012345678987654321234543210123456765432101234543212345678987654321012345678987654321 (Digit Length: 101)
19091502720519091  Palindromic Prime: 12345678987654321012345678987654321234543210123456765432101234543212345678987654321012345678987654321 (Digit Length: 101)
19091747174719091  Palindromic Prime: 123456789876543210123456789876543212345676545676543212345676545676543212345678987654321012345678987654321 (Digit Length: 105)
19091942624919091  Palindromic Prime: 12345678987654321012345678987654321234567898765432345654323456789876543212345678987654321012345678987654321 (Digit Length: 107)
19094949594949091  Palindromic Prime: 12345678987654321012345678987654567898765456789876567898765456789876545678987654321012345678987654321 (Digit Length: 101)
19096192629169091  Palindromic Prime: 12345678987654321012345678987654321234567898765432345654323456789876543212345678987654321012345678987654321 (Digit Length: 107)
19096490709469091  Palindromic Prime: 123456789876543210123456789876545678987654321012345676543210123456789876545678987654321012345678987654321 (Digit Length: 105)
19097150705179091  Palindromic Prime: 12345678987654321012345678987654321234543210123456765432101234543212345678987654321012345678987654321 (Digit Length: 101)
19108089198080191  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19108409190480191  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19109150705190191  Palindromic Prime: 12345678987654321012345678987654321234543210123456765432101234543212345678987654321012345678987654321 (Digit Length: 101)
19127290909272191  Palindromic Prime: 123456789876543212345676543234567898765432101234567898765432101234567898765432345676543212345678987654321 (Digit Length: 105)
19138092829083191  Palindromic Prime: 12345678987654321234567876543210123456789876543234567876543234567898765432101234567876543212345678987654321 (Digit Length: 107)
19149191719194191  Palindromic Prime: 1234567898765432123456789876543212345678987654321234567654321234567898765432123456789876543212345678987654321 (Digit Length: 109)
19166909090966191  Palindromic Prime: 123456789876543212345667898765432101234567898765432101234567898765432101234567898766543212345678987654321 (Digit Length: 105)
19172950905927191  Palindromic Prime: 123456789876543212345676543234567898765432101234567898765432101234567898765432345676543212345678987654321 (Digit Length: 105)
19174829092847191  Palindromic Prime: 1234567898765432123456765456787654323456789876543210123456789876543234567876545676543212345678987654321 (Digit Length: 103)
19178092829087191  Palindromic Prime: 12345678987654321234567876543210123456789876543234567876543234567898765432101234567876543212345678987654321 (Digit Length: 107)
19179072727097191  Palindromic Prime: 12345678987654321234567898765432101234567654323456765432345676543210123456789876543212345678987654321 (Digit Length: 101)
19180492829408191  Palindromic Prime: 12345678987654321234567876543210123456789876543234567876543234567898765432101234567876543212345678987654321 (Digit Length: 107)
19180926862908191  Palindromic Prime: 12345678987654321234567876543210123456789876543234567876543234567898765432101234567876543212345678987654321 (Digit Length: 107)
19180947174908191  Palindromic Prime: 1234567898765432123456787654321012345678987654567654321234567654567898765432101234567876543212345678987654321 (Digit Length: 109)
19182815151828191  Palindromic Prime: 12345678987654321234567876543234567876543212345432123454321234567876543234567876543212345678987654321 (Digit Length: 101)
19183092829038191  Palindromic Prime: 12345678987654321234567876543210123456789876543234567876543234567898765432101234567876543212345678987654321 (Digit Length: 107)
19190172727109191  Palindromic Prime: 12345678987654321234567898765432101234567654323456765432345676543210123456789876543212345678987654321 (Digit Length: 101)
19190329292309191  Palindromic Prime: 1234567898765432123456789876543210123234567898765432345678987654323210123456789876543212345678987654321 (Digit Length: 103)
19191491719419191  Palindromic Prime: 1234567898765432123456789876543212345678987654321234567654321234567898765432123456789876543212345678987654321 (Digit Length: 109)
19191682928619191  Palindromic Prime: 123456789876543212345678987654321234567876543234567898765432345678765432123456789876543212345678987654321 (Digit Length: 105)
19191826962819191  Palindromic Prime: 123456789876543212345678987654321234567876543234567898765432345678765432123456789876543212345678987654321 (Digit Length: 105)
19191891719819191  Palindromic Prime: 1234567898765432123456789876543212345678987654321234567654321234567898765432123456789876543212345678987654321 (Digit Length: 109)
19191915751919191  Palindromic Prime: 1234567898765432123456789876543212345678987654321234567654321234567898765432123456789876543212345678987654321 (Digit Length: 109)
19194172927149191  Palindromic Prime: 12345678987654321234567898765432123456765432345678987654323456765432123456789876543212345678987654321 (Digit Length: 101)
19194182928149191  Palindromic Prime: 123456789876543212345678987654321234567876543234567898765432345678765432123456789876543212345678987654321 (Digit Length: 105)
19209150705190291  Palindromic Prime: 12345678987654321012345678987654321234543210123456765432101234543212345678987654321012345678987654321 (Digit Length: 101)
19270817171807291  Palindromic Prime: 1234567898765432345676543210123456787654321234567654321234567654321234567876543210123456765432345678987654321 (Digit Length: 109)
19280828082808291  Palindromic Prime: 1234567898765432345678765432101234567876543234567876543210123456787654323456787654321012345678765432345678987654321 (Digit Length: 115)
19281529092518291  Palindromic Prime: 1234567898765432345678765432123454323456789876543210123456789876543234543212345678765432345678987654321 (Digit Length: 103)
19281708580718291  Palindromic Prime: 123456789876543234567876543212345676543210123456787656787654321012345676543212345678765432345678987654321 (Digit Length: 105)
19282806160828291  Palindromic Prime: 123456789876543234567876543234567876543210123456543212345654321012345678765432345678765432345678987654321 (Digit Length: 105)
19288070707088291  Palindromic Prime: 1234567898765432345678876543210123456765432101234567654321012345676543210123456788765432345678987654321 (Digit Length: 103)
19297919191979291  Palindromic Prime: 12345678987654323456789878987654321234567898765432123456789876543212345678987898765432345678987654321 (Digit Length: 101)
19300809190800391  Palindromic Prime: 1234567898765432100123456787654321012345678987654321234567898765432101234567876543210012345678987654321 (Digit Length: 103)
19308059195080391  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19308089198080391  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19308093139080391  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19317290909271391  Palindromic Prime: 123456789876543212345676543234567898765432101234567898765432101234567898765432345676543212345678987654321 (Digit Length: 105)
19319072727091391  Palindromic Prime: 12345678987654321234567898765432101234567654323456765432345676543210123456789876543212345678987654321 (Digit Length: 101)
19369290709296391  Palindromic Prime: 12345678987654345678987654323456789876543210123456765432101234567898765432345678987654345678987654321 (Digit Length: 101)
19371919691917391  Palindromic Prime: 1234567898765434567654321234567898765432123456789876789876543212345678987654321234567654345678987654321 (Digit Length: 103)
19379180708197391  Palindromic Prime: 12345678987654345678987654321234567876543210123456765432101234567876543212345678987654345678987654321 (Digit Length: 101)
19380939593908391  Palindromic Prime: 123456789876543456787654321012345678987654345678987656789876543456789876543210123456787654345678987654321 (Digit Length: 105)
19389180708198391  Palindromic Prime: 12345678987654345678987654321234567876543210123456765432101234567876543212345678987654345678987654321 (Digit Length: 101)
19391608580619391  Palindromic Prime: 12345678987654345678987654321234565432101234567876567876543210123456543212345678987654345678987654321 (Digit Length: 101)
19391802720819391  Palindromic Prime: 12345678987654345678987654321234567876543210123456765432101234567876543212345678987654345678987654321 (Digit Length: 101)
19391805750819391  Palindromic Prime: 12345678987654345678987654321234567876543210123456765432101234567876543212345678987654345678987654321 (Digit Length: 101)
19391870707819391  Palindromic Prime: 12345678987654345678987654321234567876543210123456765432101234567876543212345678987654345678987654321 (Digit Length: 101)
19392905750929391  Palindromic Prime: 12345678987654345678987654323456789876543210123456765432101234567898765432345678987654345678987654321 (Digit Length: 101)
19392980708929391  Palindromic Prime: 12345678987654345678987654323456789876543210123456765432101234567898765432345678987654345678987654321 (Digit Length: 101)
19395180708159391  Palindromic Prime: 12345678987654345678987654321234567876543210123456765432101234567876543212345678987654345678987654321 (Digit Length: 101)
19396290709269391  Palindromic Prime: 12345678987654345678987654323456789876543210123456765432101234567898765432345678987654345678987654321 (Digit Length: 101)
19402809190820491  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19405809190850491  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19408029192080491  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19408409190480491  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19408509190580491  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19506809190860591  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19539290709293591  Palindromic Prime: 12345678987654345678987654323456789876543210123456765432101234567898765432345678987654345678987654321 (Digit Length: 101)
19617290909271691  Palindromic Prime: 123456789876543212345676543234567898765432101234567898765432101234567898765432345676543212345678987654321 (Digit Length: 105)
19639290709293691  Palindromic Prime: 12345678987654345678987654323456789876543210123456765432101234567898765432345678987654345678987654321 (Digit Length: 101)
19705809190850791  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19708092129080791  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19719072727091791  Palindromic Prime: 12345678987654321234567898765432101234567654323456765432345676543210123456789876543212345678987654321 (Digit Length: 101)
19720809190802791  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19803809190830891  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)
19806080608060891  Palindromic Prime: 1234567898765432101234565432101234567876543210123456543210123456787654321012345654321012345678987654321 (Digit Length: 103)
19808093139080891  Palindromic Prime: 12345678987654321012345678765432101234567898765432123456789876543210123456787654321012345678987654321 (Digit Length: 101)

 

***

Michael wrote:

Q1. The first twenty such pairs p, q are

353     34543
373     345676543
383     34567876543
11311   1123211
16561   1234565654321
30803   32101234567876543210123
34843   34567876543
35053   345432101234543
35753   345676543
36563   345656543
38083   345678765432101234567876543
38183   3456787654321234567876543
38783   3456787876543
1317131 123212345676543212321
1328231 1232345678765432321
1565651 1234565654321
1597951 123456789878987654321
1987891 123456789878987654321
3064603 32101234565456543210123
3252523 323454323454323

Q2. Searching all palindromic primes p with <= 19 digits, the largest q I found had 151 digits:

q = 987654321234567898765432101234567876543210123456789876543210123456789876543234567898765432101234
5678987654321012345678765432101234567898765432123456789
p = 9190809092909080919

 

For Q2, here is a q with 153 digits arising from a p with 20 digits:

q = 123456787654321012345676543212345678987654321012345678765432123456789876543212345678987654321234567
876543210123456789876543212345676543210123456787654321

p = 180719081919180917081

***

Emmanuel wrote:

Q1
This are the first 20 :
p q
353 34543
373 345676543
383 34567876543
11311 1123211
16561 1234565654321
30803 32101234567876543210123
34843 34567876543
35053 345432101234543
35753 345676543
36563 345656543
38083 345678765432101234567876543
38183 3456787654321234567876543
38783 3456787876543
1317131 123212345676543212321
1328231 1232345678765432321
1565651 1234565654321
1597951 123456789878987654321
1987891 123456789878987654321
3064603 32101234565456543210123
3252523 323454323454323

Q2
This is my biggest  q :
9876543210123456789876543210123456789...123456789876543210123456789 (7427 digits)

p=909090...3482843...090909 (831 digits)

***

Jeff wrote.

Q1. I believe the next 20 pairs are:

11311    1123211
16561    1234565654321
30803    32101234567876543210123
34843    34567876543
35053    345432101234543
35753    345676543
36563    345656543
38083    345678765432101234567876543
38183    3456787654321234567876543
38783    3456787876543
1317131    123212345676543212321
1328231    1232345678765432321
1565651    1234565654321
1597951    123456789878987654321
1987891    123456789878987654321
3064603    32101234565456543210123
3252523    323454323454323
3331333    333212333
3487843    3456787876543
3515153    345432123454321234543

 

Q2. p = 309333727333903
      q = 32101234567898765433345676543234567654333456789876543210123 (59 digits)

***

Giorgos wrote:

Q1, the first 20 pairs p->q are: 

353->34543
373->345676543
383->34567876543
16561->1234565654321
30803->32101234567876543210123
38083->345678765432101234567876543
3709073->34567654321012345678987654321012345676543
7093907->765432101234567898765434567898765432101234567
191969191->1234567898765432123456789876789876543212345678987654321
11918381911->11234567898765432123456787654345678765432123456789876543211
35180908153->3454321234567876543210123456789876543210123456787654321234543
37092829073->3456765432101234567898765432345678765432345678987654321012345676543
39061816093->3456789876543210123456543212345678765432123456543210123456789876543
90619391609->987654321012345654321234567898765434567898765432123456543210123456789
1080826280801->10123456787654321012345678765432345654323456787654321012345678765432101
1080919190801->1012345678765432101234567898765432123456789876543212345678987654321012345678765432101
3919292929193->345678987654321234567898765432345678987654323456789876543234567898765432123456789876543
109290838092901->101234567898765432345678987654321012345678765434567876543210123456789876543234567898765432101
117090818090711->11234567654321012345678987654321012345678765432123456787654321012345678987654321012345676543211
119191818191911->11234567898765432123456789876543212345678765432123456787654321234567898765432123456789876543211

Q2. First I tried to find p as small as possible in order to get q titan palprime

p = 909090909090909090909090909090909090909090909090909209009171900902909090909090909090909090909090909090909090909090909 (117 digits)

gives a palprime q with 1003 digits (we can easily find q from p so I will not send it here)

Then I tried to find q with more than 10000 digits

p=9090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090
909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090
909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090
909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090
909090909090909090909090909090909090909090909090909091718612168171909090909090909090909090909090909090909090909090909090909090
909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090
909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090
909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090
909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909  (1123 digits)

gives a palprime q with 10043 digits

***

Simon wrote:

Q1
p=11, q=11
p=101, q=101
p=353, q=34543
p=373, q=345676543
p=383, q=34567876543
p=787, q=787
p=11311, q=1123211
p=16561, q=1234565654321
p=30803, q=32101234567876543210123
p=32323, q=32323
p=34543, q=34543
p=34843, q=34567876543
p=35053, q=345432101234543
p=35753, q=345676543
p=36563, q=345656543
p=38083, q=345678765432101234567876543
p=38183, q=3456787654321234567876543
p=38783, q=3456787876543
p=76667, q=76667
p=78787, q=78787

Q2
The largest p = q palprime is 111..111, the largest known prime RepUnit 8177207 digits long.
See https://www.elektrosoft.it/matematica/repunit/repunit.htm

The largest p = q palprime I could find myself:
Parallel processing happend to find two, both 75941 digits long.
Replace each .. with 37955 more zero's:
p = q = 110..0110101100100010011010110..011
p = q = 110..0110101001100011001010110..011

***