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.

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

***
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

***

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.

***

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

***

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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