Problems & Puzzles: Puzzles

Puzzle 687 17769643

From the Prime Curio pages we have this nice finding:

17769643 This prime, its predecessor, and its successor form the first trio that concatenates all 6 ways to form primes [Merickel]

Q1. Find some more primes like 17769643
Q2. Find the least prime for a quartet like that
(or the best qualified quartet).


Contributions came from Jeff Heleen, Felipe I. G., J. K. Andersen, Emmanuel Vantieghem, Giovanni Resta, W. E. Clark & Jan van Delden 

***

Jeff wrote

There are 7 other sets of primes <10^9 which satisfy the
condition for Q1:

65918767, 65918777 and 65918779
98182393, 98182439 and 98192453
199387411, 199387417 and 199387421
253802201, 253802257 and 253802281
279011897, 279011899 and 279011921
712229239, 712229257 and 712229267
745427659, 745427663 and 745427689

***

Felipe wrote:

Hola Carlos,
Algunos resultados a Q1:
17.769.629 17.769.643 17.769.677
65.918.767 65.918.777 65.918.779
98.182.393 98.182.439 98.182.453
199.387.411 199.387.417 199.387.421
253.802.201 253.802.257 253.802.281
279.011.897 279.011.899 279.011.92

***

Andersen wrote

Q1. The first such primes are: 17769643, 65918777, 98182439, 199387417,
253802257, 279011899, 712229257, 745427663, 1393567283, 1616839351,
1631014841, 1907708203, 1964034269, 1974494453, 2453507809, 2942863481.
 
Q2. There are n! ways to concatenate n numbers, so a quartet has 24
concatenations. This is computationally infeasible.
{313, 317, 331, 337} gives 10 primes out of 24 numbers. The smallest
improvement is 11 for {540716711, 540716747, 540716753, 540716777
}.

***

Emmanuel wrote

1. All primes with Merickels property below  10^9  are :
              17769643, 65918777, 98182439, 199387417, 253802257, 279011899, 712229257, 745427663.
       I think there could be infinitely many.
 
Q2. I think such quartets do not exist.
       If we take four primes  p, q, r, s  'ad random' and form their 24 concatenations, the maximum number of primes obtained perhaps is  16
       (when p, q, r, s = 3, 71, 223, 8969  or  7, 97, 709, 3217   and maybe a few others).

***

Giovanni wrote

The solution of Q1 was easy (I quickly found several such primes) but I deleted the file by accident. But surely you'll have other contributions from other people.

For Q2 I searched all the quartets up to 238*10^9 to find
the ones which produce most primes (max possible is 24, since there are
24 permutations of the 4 consecutive primes).

The best I found was 11 primes for the quartet starting at 540716711.
The next quartet with 11 primes starts at 100011200603 (quite a gap).
I found several other such quartets, but none producing 12 or more primes (until now).

...

For the sake of completeness I rewrote and rerun the program for Q1, the  triples, at least for a while.

So the first triples are centered at 17769643, 65918777, 98182439, 199387417,... and the 100th triple is centered at 90366876511.

I've also searched for the first triple of consecutive primes A,B,C such
that the 6 numbers AB, BA, AC, CA, BC, CB are primes.
The first such triples are {44299781, 44299823, 44299859} and
{137233013, 137233049, 137233073}.

...

I have found the earliest quadruple of consecutive primes which gives 12 primes (half of the 24 permutations).
It is 496653276961, 496653276989, 496653277019, 496653277037.
 

***

Clark wrote

I made no progress on Puzzle 687 but it did prod me to submit
a new sequence to the OEIS, namely, primes p such that if q
is the next prime then the concatenation of p with q and the
concatenation of q with p are both primes. The first few terms are:

This is related to sequence https://oeis.org/A030459 which is
primes p such the the concatenation of p with the next prime
is prime.
 

***

Jan wrote

Solutions to Q1:
 
  17769629,  17769643,  17769677  
  65918767,  65918777,  65918779 
  98182393,  98182439,  98182453  
199387411, 199387417, 199387421
253802201, 253802257, 253802281 
279011897, 279011899, 279011921
712229239, 712229257, 712229267
745427659, 745427663, 745427689
1393567271,1393567283,1393567291
1616839337,1616839351,1616839361
1631014793,1631014841,1631014849
1907708171,1907708203,1907708207
1964034221,1964034269,1964034271
1974494449,1974494453,1974494471
2453507807,2453507809,2453507821
2942863463,2942863481,2942863549
3532205351,3532205357,3532205371
3558773201,3558773209,3558773221
3579874729,3579874747,3579874757
4364416241,4364416249,4364416271
4772986357,4772986459,4772986469
4930999883,4930999897,4930999921
4989797903,4989797909,4989797941
5284526167,5284526201,5284526209
5492793707,5492793751,5492793781
6185269459,6185269469,6185269501
6221570521,6221570549,6221570557
6474195919,6474195929,6474195947
6688249423,6688249427,6688249433
6760616281,6760616299,6760616339
 
Q2:
 
I did test untill 6.10^9 and found no solutions. A relatively small subrange shows that itís rather hard to obtain more than 10 concatenations that are prime out of the 24 required.
Solutions will probably exist, but will be large.

***

On May 19, Abhiram R Devesh wrote:

Please find my submissions to puzzle 687 Q2. Here I have considered alternating primes that can permutate to form at least 10 primes

 366199   366217   366227    366259 
 453137   453157   453181    453199 
 567607   567649   567659    567667 
 579611   579629   579641    579653 
5306659  5306683  5306701   5306713 
5456597  5456663  5456699   5456707

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