Problems & Puzzles: Puzzles

Puzzle 1054. q=concat(x,y) Paolo Lava sent the following nice puzzle. Consider a prime q=concat(x,y) and take the sum k of p(x)+p(y) for any possible x and y. If k is prime repeat the process with k. Which are the least primes that through this process reach another prime for n consecutive times? I found q for n<5: n=1  q=13        k = p(1)+p(3) = 2 + 5 = 7 (prime) n=2  q=271      k = p(2)+p(71) + p(27)+p(1) = 3+353 + 103+2 = 461 (prime)                       k = p(4)+p(61) + p(46)+p(1) = 7+283 + 199+2 = 491 (prime) n=3  q=331      k = p(3)+p(31) + p(33)+p(1) = 5+127 + 137+2 = 271 (prime)                        k = p(2)+p(71) + p(27)+p(1) = 3+353+ 103+2 = 461 (prime)                        k =p(4)+p(61) + p(46)+p(1) = 7+283 + 199+2 = 491 (prime) n=4  q=98911   k = p(9)+p(8911) + p(98)+p(911) + p(989)+p(11) + p(9891)+p(1) = 23+92251 + 521+7109 + 7823+31 + 103511+2 = 211271 (prime)                        k = p(2)+p(11271) + p(21)+p(1271) + p(211)+p(271) + p(2112)+p(71) + p(21127)+p(1) = 3+119687 + 73+10357 + 1297+1741 + 18439+353 + 238829+2 = 390781 (prime)                        k = p(3)+p(90781) + p(39)+p(0781) + p(390)+p(781) + p(3907)+p(81) + p(39078)+p(1) = 5+1170167 + 167+5953 + 2687+5953 + 36847+419 + 468113+2 = 1690313 (prime)                        k = p(1)+p(690313) + p(16)+p(90313) + p(169)+p(0313) + p(1690)+p(313) + p(16903)+p(13) + p(169031)+p(3) = 2+10414939 + 53+1163629 + 1009+2081 + 14419+2081 + 186773+41 + 2292947+5 = 14077979 (prime) Q. Find the minimal q for n>5?

---

During the two weks from Set 4 to Set 17, contributions came from Giorgos Kalogeropoulos, Emmanuel Vantieghem, Oscar Volpatti

***

Giorgos wrote:

n=5  q=335185321

k = p(3) + p(35185321) + p(33) + p(5185321) + p(335) + p(185321) + p(3351) + p(85321) + p(33518) + p(5321) + p(335185) +p(321)  + p(3351853) + p(21)  p(33518532) +p(1) = 

5 + 678276667 + 137 + 89416781 + 2251 + 2532853 + 31121 + 1094473 + 395533 + 52021 + 4796119 + 2131 + 56239583 + 73 + 644425829 + 2 = 1477265579 (prime)  
 

k = p(1) + p(477265579) + ....+ p(147726557) + p(9) = 2 + 10512101609 + ... + 3071545493 +  23 = 15566693003 (prime)

 

k = ...= 190750163243 (prime)

 

k= ... = 3073872299231 (prime)

 

k= ... = 13883772743609 (prime)   

***

Emmanuel wrote:

Starting with  335185321  Paoio's process generates five primes : 

   1477265579, 15566693003, 190750163243, 3073872299231, 13883772743609.
    (the next step gives  197144899865523  which is not prime

I'm afraid the next step will involve numbers  n  that are too big to find the n-th prime in a reasonable time..

***

Oscar wrote:

I gave a try to puzzle 1054, but I couldn't search too far.

These are the first ten solutions with n = 4:

 

98911...14077979
1445959...853243471
1963567...17320218479
5496571...23827600969
5958871...8376784489
14406097...19990648889
15621901...15571634269
16591301...165037770077
17464619...253184172403
24643621...14199045053

No solutions with n > 4 found so far...

...

I finally found the minimal prime q = 335185321 for n = 5:  
335185321, prime
1477265579, prime
15566693003, prime
190750163243, prime
3073872299231, prime
13883772743609, prime
197144899865523, composite

Here below the primes used within last step (unsuccessful).
p(1388377274360) = 42117893830253
p(138837727436) = 3880265914039
p(13883772743) = 354758411269
p(1388377274) = 32134928773
p(138837727) = 2877639223
p(13883772) = 253955263
p(1388377) = 21985189
p(138837) = 1854247
p(13883) = 150383
p(1388) = 11503
p(138) = 787
p(13) = 41
p(1) = 2
p(3883772743609) = 121952448341473
p(883772743609) = 26396726645561
p(83772743609) = 2297320326121
p(3772743609) = 91269405277
p(772743609) = 17411028409
p(72743609) = 1458148841
p(2743609) = 45446129
p(743609) = 11277439
p(43609) = 527069
p(3609) = 33703
p(609) = 4483
p(09) = 23
p(9) = 23

***