Problems & Puzzles: Collection 20th

Coll.20th-008. Primes of the migrant digit

On March 16 2018, Jaime Ayala wrote:

Find the largest prime P of length L such that all the integers obtained by migration of the leftmost digit to each of the righter positions are distinct primes, being all the primes of the same length L.

Examples given by Mr. Ayala

L=2, Primes: 13->31

L=3, Primes: 149->419 ->491

L=6, 984427 -> 894427 -> 849427 -> 844927 -> 844297-> 844279

Carlos Rivera found this example and the largest below 2^32:

L=10, Primes: {3286760681->2386760681->2836760681->2863760681->2867360681->2867630681->2867603681->2867606381->2867606831->2867606813}

Q. Send your largest one example.


Contributions came from Jan van Delden


On July 19, 2018, Jan wrote:

11 digits: 71043402931

12 digits: 352968841549, 712845952529

13 digits: No solution


Admissible numbers:


First digit and last digit in {1,3,7,9}.
Second digit in {1..9}. 
Other digits in {0..9}.
All digits, except last one, unequal to the last digit.


Constructing admissible numbers first (with a fixed length) and testing for prime later has several advantages if the length becomes large. Sieving as a test for being prime of length L is not an option anymore. If the length L is larger than 20 the number of generated admissible numbers is smaller than the number of primes. But more importantly one could add a test mod 3 and a test mod 11 in the construction process to determine if any of the resulting migrated numbers are divisible by 3 or 11, before actually computing these migrated numbers. For divisor 3 this is just 1 test for all migrated numbers, but for 11 this test is specific for each migrated number (at the cost of precomputing L different sums mod 11).

An alternative could be to construct the L migrated numbers in one go. A disadvantage could be that this would entail unneccesary constructing numbers that are divisible by 11. So I didnít pursue this. [If we assume each migrated number is divisible by 11 with probability p, the probability of at least 1 migrated number being divisible by 11 would be 1-(1-p)^L, which tends to 1 with increasing L].



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