Problems & Puzzles: Puzzles

 Puzzle 850. The wagon prime. Here we ask for prime numbers P, such that every k consecutive digits inside P is a distinct prime number pi. Example: For k=2, P=411379717319 is a prime number, and every two consecutive digits inside P is a distinct prime number: 41, 11, 13, 37, 79, 97, 71, 17, 73, 31 & 19. Moreover, 411379717319 is the earliest prime number producing 11 two consecutive digits prime numbers this way and no more than eleven may be produced this way. I think of this kind of primes as wagons. See why... the eleven two digits primes seems to be traveling as a wagon on the rails of P... 411379717319 411379717319 411379717319 411379717319 411379717319 411379717319 411379717319 411379717319 411379717319 411379717319 411379717319 P=1061979199733313739311933719319137 is the prime number for k=4, equivalent to 411379717319, because 1061979199733313739311933719319137 is the earliest prime number producing 31 four consecutive digits prime numbers this way and no more than 31 may be produced this way. Q1. Can you confirm or disconfirm my claims for k=2 &4? Q2. Can you get the equivalent solutions for k=3, 5, 6, ... 10?

Contributions came from Jan van Delden, Emmanuel Vantieghem, Shyam Sunder Gupta and Michael Hürter

***

Jan wrote:

k  wagons rail
2  11     411379717319
3  26     1019113137977397199193733179
4  31     1061979199733313739311933719319137
5  47     374831379939791939113997931991393133317939371999713
6  29     1971473911777911173191173713117119
7  23     24071479791317917331771937311
8  24     3043843379977173139111931991779
9  24     46140475173113373193179799197719
10  24     795318529179717731117371197179719

The wagons belonging to k=3,4,7,8,9 contain a 0.

Solutions without a 0 are:

k wagons rail
3 26     2419113137971991937397733173
4 31     1451979199733313739311933719319133
7 23     51677877797191131131113193717

8 22     31533563377137139771331131733
9 23     3896945771133799977137937971317

***

Emmanuel wrote:

Q1. Confirmed !

Q2.
k        p                                                                                        m
1     2357                                                                                     4
3     1019113137977397199193733179                                           26
5     374831379939791939113997931991393133317939371999713   47
6     158747391177791117319117371311711                                    28
7     24071479791317917331771937311                                          23
8     3043843379977173139111931991779                                       24

(k = number of 'moving' digits, p = least prime, m = maximum number of primes that can be obtained that way)

***

Shyam wrote:

1) Your Claim for K=2 and 4 is confirmed. Additionally number of primes including smallest and largest are given in Answer to Q 2
2) Due to limited time available I could find all values for K=2 to 7. The details are as follows:
k=2: Max Number of distinct primes possible =11
The smallest prime Pmin=411379717319
The Largest Prime Pmax = 619737131179
All primes producing 11 two consecutive digits prime numbers are given below:
411379717319,411737131979,413119737179,413197371179,413717311979,413719731179,413797317119,417113797319,
417371131979,417973711319,419711317379,419713731179,419731371179,611373171979,611719731379,611731971379,
611971731379,613117973719,
613173711979,613197371179,617131197379,617137973119,617311371979,617311971379,
617313711979,619737131179 (Total 26)

k=3: Max Number of distinct primes possible =26
The smallest prime Pmin= 1019113137977397199193733179
The Largest Prime Pmax = 9419919379719113739773313173
All primes producing 26 three consecutive digits prime numbers are given below:
1019113137977397199193733179,1019113173313797739719919373,1019113173977331379719919373,
1019193797199113739773313173,
1019193797733173971991131373,1019373317397191991131379773,
1019373977331317971991911379,
1019379719919113173313739773,1019379773313173971991911373,
1019379773971991911313733179,
1019911313739719193797733173,1019911317977397191937331379,
1019911373313173971919379773,1019911379719193739773313173,1019911379773313173971919373,
1019919113173313797739719373,
1019919113733131739719379773,1019919113797193739773313179,
1019919113797733131739719373,1019919373313173971911379773,1019919379719113173313739773,
1019919379773971911373313179,
2419113137971991937397733173,2419113179773313739719919379,
2419193797733173971991131373,2419193797739719911313733173,2419199113137331797739719379,
2419199113173313797739719373,
2419199113739719379773313173,2419373313179773971919911379,
2419373977331317971919911379,2419379719199113739773313173,2419379719919113733131739773,
2419379773313739719919113179,2419379773971919911373313179,2419911317397733137971919373,
2419911373977331317971919379,
2419919373977331379719113173,2419919379773971911313733173,
4019113137977397199193733173,4019193739773313179719911379,4019193797739719911373313173,
4019193797739719911373313179,
4019373971919911317331379773,4019379719199113739773313173,
4019379773971919911313733173,4019379773971991911373313173,4019911313797739719193733179,
4019919113137971937397733179,4019919113733131739719379773,4019919379773313739719113179,
4619199113137977331739719373,
4619199113179719373977331379,4619199113797193739773313173,
4619911317397733137971919373,
4619911379773313173971919373,4619911379773971919373313173,
4619919113137971937331739773,
4619919373313797739719113179,4619919379773313739719113179,
4619919379773971911373313173,5419113137971991937397733179,5419113797199193733131739773,
5419113797199193739773313173,
5419193739719911313797733173,5419193797733137397199113179,
5419199113739719379773313179,
5419373313797739719919113173,5419373313797739719919113179,
5419373971919911379773313179,
5419379773971919911313733173,5419911313733173971919379773,
5419911317331373971919379773,
5419919113173313739719379773,5419919373313179773971911379,
5419919373971911313797733179,
5419919379773971911313733179,6019113137977397199193733173,
6019113733131739719919379773,
6019193797739719911313733179,6019199113739773313179719379,
6019373313179773971919911379,6019373971919911379773313173,6019373971991911313797733179,
6019911313733179773971919379,
6019911379719193739773313179,6419113173313797739719919373,
6419113739719919379773313173,6419113739719919379773313179,6419199113137971937397733179,
6419199113797739719373313179,
6419373313173971991911379773,6419911317397191937977331373,
6419911317397733137971919373,6419911317971919373977331379,6419919113137977397193733173,
6419919113173971937977331373,6419919379719113739773313173,6619113137331797739719919379,
6619113137971991937397733173,6619113739719919379773313179,6619193733131739719911379773,
6619193797739719911317331373,6619373313179773971919911379,6619379719199113137397733179,
6619379719919113733131739773,
6619379773313739719919113173,6619379773971991911373313179,
6619911313739773317971919379,
6619919113137397193797733173,6619919113137971937397733179,
6619919113739773313179719379,6619919373971911379773313179,6619919379773313739719113173,
7019113137397733179719919379,7019193733131739719911379773,7019193739719911317331379773,
7019193739773313179719911379,
7019199113137977331739719373,7019199113179773313739719379,
7019373313797191991131739773,
7019373313797739719199113173,7019373317977397191991131379,
7019373971991911313797733173,
7019379773313739719919113179,7019379773317397199191131373,
7019911313797739719193733173,
7019911379773971919373313179,7019919113137971937397733179,
7019919373313797739719113173,
7019919373971911313797733173,7019919379719113173313739773,
7019919379719113173977331373,
7619113739773313179719919379,7619193733179773971991131379,
7619193739773313179719911379,
7619199113137397193797733173,7619199113137977397193733179,
7619199113797193739773313179,
7619373971919911379773313179,7619373971991911317977331379,
7619373977331317971919911379,
7619373977331379719199113173,7619373977331797199191131379,
7619379719199113739773313173,
7619379719919113739773313179,7619379773971919911313733173,
7619911313797191937397733179,
7619911373977331317971919379,7619919373977331317971911379,
9419113179773971991937331379,
9419113797733131739719919373,9419193733137977397199113179,
9419193797199113137397733173,9419199113179773313739719379,9419199113797193733131739773,
9419199113797193739773313173,9419373971991911379773313173,9419379719199113137331739773,
9419379719919113137331739773,
9419911313739719193797733173,9419911313797191937331739773,
9419919113137971937397733179,9419919113797739719373313179,9419919373971911317977331379,
9419919379719113739773313173(Total 166).

k=4: Max Number of distinct primes possible =31
The smallest prime Pmin= 1061979199733313739311933719319137
The Largest Prime Pmax = 9551979199733313739311933719319133
All primes producing 31 four consecutive digits prime numbers are given below:
1061979199733313739311933719319137,1451979199733313739311933719319133,3301979199739311933719319137333137,
370197919973931193371931913733
3137,4001979199739311933719319137333137,7001979199733313739311933719319133,
955197919973331373931193371931
9133(Total 7)

k=5: Max Number of distinct primes possible =47
The smallest prime Pmin= 374831379939791939113997931991393133317939371999713
The Largest Prime Pmax = 9551979199733313739311933719319133
All primes producing 47 five consecutive digits prime numbers are given below:
374831379939791939113997931991393133317939371999713, 763031379939791939113997931991393133317939371999719(Total 2)

k=6: Max Number of distinct primes possible =29
The smallest prime Pmin= 1971473911777911173191173713117119
The Largest Prime Pmax = 9651473911777911173191173713117119
All primes producing 29 six consecutive digits prime numbers are given below:
1971473911777911173191173713117119,9651473911777911173191173713117119(Total 2)

k=7: Max Number of distinct primes possible = 23
The smallest prime Pmin= 24071479791317917331771937311
The Largest Prime Pmax = 99071479791317917331771937311
All primes producing 23 seven consecutive digits prime numbers are given below:
24071479791317917331771937311,40675877797191131131113137977,51677877797191131131113193717,
70435577797191131131113193717,
70777877797191131131113137977,70777877797191131131113193717,
77435577797191131131113137977,
91447877797191131131113193717,99071479791317917331771937311
(Total 9)

It is easy to workout for higher values of K depending upon availability of computational time.

***

Michael wrote:

Q1:
I have the same results.
Q2:
I have the following results for n= 3, 5, ... 10 :

n  result
------------------------------
--------------------------
3  1019113137977397199193733179
5  374831379939791939113997931991
393133317939371999713
6  197147391177791117319117371311
7119
7  24071479791317917331771937311
8  304384337997717313911193199177
9
9  461404751731133731931797991977
19
10 793831852917971773111737119717
9713

***

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