Problems & Puzzles: Puzzles

Puzzle 1053. p=concat(a,b) such that... Paolo Lava sent the following nice puzzle. Let p = concat(a,b) be a reversible prime with k digits such that, with all the possible k-1 concatenation of two numbers a and b, we get k-1 distinct primes (a mod b) and the sum of these k-1 primes is a reversible prime itself. I found the first ten terms: 37, 79, 97, 353, 383, 389, 37463, 79687, 313517, 3112913 Here below the process for the last two terms. 313517 and 715313 are both prime 31351 mod 7 = 5 3135 mod 17 = 7 313 mod 517 = 313 31 mod 3517 = 31 3 mod 13517 = 3 5+7+313+31+3 = 359 that is prime like 953. 3112913 and 3192113 are both prime 311291 mod 3 = 2 31129 mod 13 = 7 3112 mod 913 = 373 311 mod 2913 = 311 31 mod 12913 = 31 3 mod 112913 = 3 2+7+373+311+31+3 = 727 that is prime and palindromic. Q. Are there other terms?

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During the two weks from Set 4 to Set 17, contributions came from Oscar Volpatti, Vicente Felipe Izquierdo, Giorgos Kalogeropoulos, Simon Cavegn, Emmanuel Vantieghem

 

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Oscar wrote:

I checked all reversible primes below 10^13, finding two more terms.

 

733193633: 2+29+179+659+7331+733+73+7 = 9013.

7331702267: 5+61+149+929+773+7331+733+73+7 = 10061.

 

Note: I initially checked only zeroless reversible primes, judging decompositions like 7331702267 = concat(73317,02267) not admissible.

But I noticed that the decompositions 390781 = concat(39,0781) and 1690313 = concat(169,0313) are used within the explanation of puzzle 1054.

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Felipe wrote:

the next 2 on the list are 733193633 and 7331702267

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Giorgos wrote:

Next term is 733193633

 

733193633 and 336391337 are both primes
 

 

73319363 mod 3 = 2
 

7331936 mod 33 = 29
 

733193 mod 633 = 179
 

73319 mod 3633 = 659
 

7331 mod 93633 = 7331
 

733 mod 193633 = 733
 

73 mod 3193633 = 73
 

7 mod 33193633 = 7 

 

2+29+179+659+7331+733+73+7 = 9013 which is prime like 3109

No more terms < 5*10^11

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Simon wrote:

733193633
7331702267

Searched up to 10^14

Without the distinct condition: Found these terms, also searched up to 10^14:
37, 79, 97, 353, 383, 389, 3203, 3803, 37463, 79687, 311137, 311957, 312289, 313517, 371057, 715109, 797009, 3112913, 3137003, 3600017, 7100003, 71947037, 380000009, 733193633, 3794000333, 7193124947, 7331702267, 37337357017

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Emmanuel wrote:

I found two bigger examples :

   733193633 is an emirp
     73319363 mod 3 = 2
     7331936 mod 33 = 29
     733193 mod 633 = 179
     73319 mod 3633 = 659
     7331 mod 93633 = 7331
     733 mod 193633 = 733
     73 mod 3193633 = 73
     7 mod 33193633 = 7
     and  2 + 29 + 179 + 659 + 7331 + 733 + 73 + 7 = 9013  is an emirp

   7331702267 is an emirp
     733170226 mod 7 = 5
     73317022 mod 67 = 61
     7331702 mod 267 = 149
     733170 mod 2267 = 929
     73317 mod 02267 = 773
     7331 mod 702267 = 7331
     733 mod 1702267 = 733
     73 mod 31702267 = 73
     7 mod 331702267 = 7
     and  5, 61, 149, 929, 773, 7331, 733, 73, 7 = 10061  is an emirp

I think that there are no more examples.

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