Problems & Puzzles: Puzzles

Puzzle 587. Concatenating absolute differences Based on the puzzle 662 of the blog on numbers by Claudio Meller I, have constructed a similar game. Given an integer N1 of D digits you generate an integer N2 of (D-1) digits concatenating orderly the absolute differences of contiguous digits of N1. Proceed recursively from N1 to generate N2, N3, ..., Nk. Example: 9175 -> 862 -> 24 -> 2 Zeros by the left generated in a step, are discarded for the next step. Our puzzle consists in getting sequences of integers  N1, N2, ..., Nk such that all of them are prime integers. My largest solutions free of zeros by the left is this one for K=9: 528888881 36000007 3600007 360007 36007 3607 367 31 2 If zeros by the left are allowed in just one step, I got two solutions for K=11 656088008809 456088008809 11680808089 11680808089 528888881 528888881 36000007 36000007 3600007 3600007 360007 360007 36007 36007 3607 3607 367 367 31 31 2 2 Q1. Send you largest solution without zeros by the left Q2. Send you largest solution allowing zeros by the left in just one step.

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Contributions came from Giovanni Resta, Emmanuel Vantieghem, Jan van Delden, Hakan Summakoğlu.

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Giovanni wrote:

For Q1, I found 4 sequences of length 14.
They start at
12931520935206062288222663, 72319124519868604426880809,
92115322955440402882282063, 92337988155448402882282063 and
and the end at 1011011100101. This is one of the sequences:

12931520935206062288222663
1762432962326664060600403
614211734114002466660443
53210641303402220006401
2111623233142002006241
100541110232202206423
10513001211022026221
1542301110120224401
412131001112202041
31122101001022243
2010111101120021
211100011012021
10010010111221
1011011100101

For Q2 I found several sequences of length 16. One of them:

13840006750000606000026847
2544006125000666600024243
310406513500600060022221
21446142250660066020001
1302532035606060622001
232321232166666640201
11111111150000024221
45000022201
1500020021
450022021
15020221
4522201
130021
23021
1321
211

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Emmanuel wrote:

Q1. I found a chain of  11  primes, without zero by the left :

1081620080800841, 187542088880843, 71212280008841, 6111068008043, 500162808841, 50154688043, 5141220841, 433102843, 10212641, 1211423, 110321.
 

Q2. allowing just one zero, I found a chain of twelve primes :

351000802048069, 24100882244863, 2310806020423, 121886622421, 11702040221, 0672244201, 15020221, 4522201, 130021, 23021, 1321, 211.

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Jan wrote:

Q1:

K=14

 

92337988155448402882282063

7104210740104442606066263

614211734114002466660443

53210641303402220006401

2111623233142002006241

100541110232202206423

10513001211022026221

1542301110120224401

412131001112202041

31122101001022243

2010111101120021

211100011012021

10010010111221

1011011100101

 

Q2:

 

K=15 (3 solutions)

 

3813690137234288866666660223

5871810917036228260606066201

5871810917036228260606066243

 

all ending with the prime 101101110101

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Hakan Summakoğlu wrote:

Q1: My largest solution is like yours. And There isn't solution for k>9. My second largest solution is for k=7. I got 8 solution for k=7. 

3282841	3282847	5066801	5606047
166643	166643	560281	166643
50021	50021	16267	50021
5021	5021	5441	5021
521	521	103	521
31	31	13	31
2	2	2	2

 

8002847	9246403	9246841	9246847
802643	722243	722243	722243
82421	50021	50021	50021
6221	5021	5021	5021
401	521	521	521
41	31	31	31
3	2	2	2

Q2: My largest solution is like yours. And There isn't solution for k>11. My second largest solution is for k=9. I got 1 solution for k=9.

3906080081
696688087
33020881
322807
10687
1621
541
13
2

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