Problems & Puzzles: Puzzles

 Puzzle 583. Traveling intertwined primes Claudio Meller sent the following nice puzzle: Given two primes, intertwine them to get two primes. Examples For n= 2: 13 & 43 1433 & 4133 For n = 3: 139 & 271 123791 & 217319   For n = 4: 1399 & 2467 12349697 & 21436979  Q1. Get larger examples Base on the Claudio's idea I simply exhausted it and posed this: Given two primes, each of length n, running in opposite direction, get 2*n distinct primes intertwining them. Example: For n=4 2129 & 4937 ->     <- 21294937 21249937 21429937 24192397 42913279 49231729 49327129 49372129 Q2. Get Larger examples

Contributions came from Torbjörn Alm, Hakan Summakoğlu, Emmanuel Vantieghem, Jan van Delden, J. K. Andersen.

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Torbjörn Alm wrote:

Q1: There are 13242 solutions for n = 4:

Lowest Solution for n=4:
1039 & 1009
11003099 & 11000399

Highest Solution for n=4:
9973 & 9697
99967937 & 99699773

There are 527908 solutions for n=5:

First Solution for n=5:
10069 & 10039
1100006399 & 1100003699

Highest Solution for n=5:
99991 & 99871
9999989711 & 9999897911

First Solution for n=6:
100151 & 100019
110000105119 & 110000011591

Highest Solution for n=6:
999029 & 998951
999998092591 & 999989905219

Highest Solution for n=7:
9999991 & 9999463
99999999949613 & 99999999496931

Highest Solution for n=8:
99999989 & 99999089
9999999999908899 & 9999999999098899

Highest Solution for n=9:
999999797 & 999999761
999999999999779671 & 999999999999776917

Q2:

There is only one more solution for n=4 with distinct primes:
Solution for: 3361 & 2437
1: 33612437
2: 33621437
3: 33264137
4: 32346317
5: 23433671
6: 24333761
7: 24337361
8: 24373361

Unfortunately it is smaller than the example. In the remaining 4 solutions, the first digit in prime #1 and the last
digit in prime #2 are the same (1), and thus the first two lines are identical.

Solution for: 29017 & 19231
1: 2901719231
2: 2901179231
3: 2901197231
4: 2910912731
5: 2199021371
6: 1299203117
7: 1922930117
8: 1922391017
9: 1923219017
10: 1923129017

I have run 6 digits up to about 600000 with no hit.

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Hakan Summakoğlu wrote:

Q1:
For n = 14:
10000000000051 & 10000000001161

1100000000000000000001015611 &
1
100000000000000000010106511

Q2: For n = 5:

19231 & 29017
1923129017
1923219017
1922391017
1922930117
1299203117
2
199021371
2910912731
2901197231
2901179231
2901719231

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Jan van Delden wrote:

Q1: Solutions with p and q each consisting of different digits (and no digit equal to 0) having different digits at the same decimal position, smallest p.q:

n           p        q
1           3        7
2          13       67
3         137      491
4        1237     2179
5       12347    25763
6      123457   254971
7     1234657  2176843
8    12345769 23469871

No solution possible with these kind of p,q for n>=9.

Q2: Solutions with smallest p.q with 2n different primes, no further conditions on p,q:

n        p     q
1        3     7
2       11    23
3      211   283
4     2129  4937
5    19231 29017
6    No solution

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Emmanuel Vantieghem wrote:

Q1.  There are 527908 sets  {p,q}  of five-digits primes  p  and  q  that intertwine to two primes.  The number of such sets increases rapidly with the number of digits  n  of the primes  p  and  q.  For  n = 5, 6, ... , 30  the smallest  p  for which there is a smallest  q  that intertwines to two primes is (I only print  p  and  q) :

10007 ,  10691

100003 ,  100183

1000003 ,  1001311

10000019 ,  10002287

100000007 ,  100000541

1000000007 ,  1000000427

10000000019 ,  10000001519

100000000003 ,  100000000069

1000000000039 ,  1000000004893

10000000000037 ,  10000000037393

100000000000031 ,  100000000002563

1000000000000037 ,  1000000000002371

10000000000000061 ,  10000000000008113

100000000000000003 ,  100000000000002307

1000000000000000003 ,  1000000000000025719

10000000000000000051 ,  10000000000000000147

100000000000000000039 ,  100000000000000027513

1000000000000000000117 ,  1000000000000000074679

10000000000000000000009 ,  10000000000000000001269

100000000000000000000117 ,  100000000000000000021453

1000000000000000000000007 ,  1000000000000000000018271

10000000000000000000000013 ,  10000000000000000000058021

100000000000000000000000067 ,  100000000000000000000058813

1000000000000000000000000103 ,  1000000000000000000000012589

10000000000000000000000000331 ,  10000000000000000000000037549

100000000000000000000000000319 ,  100000000000000000000000027763

1000000000000000000000000000057 ,  1000000000000000000000000389763

All intertwining are proved primes.  The only reason I limited my search to 30 digits is that the time to prove the primality of the intertwined numbers becomes too big.

Q2.  Here, the conditions are much stronger, so I guess that the number of solutions is finite.  I found two five digits primes  p = 19231 and  q =  29017  that give the following chain of ten primes :

1923129017, 1923219017, 1922391017, 1922930117, 1299203117, 2199021371, 2910912731, 2901197231, 2901179231, 2901719231.

This is the only solution for 5-digit primes.  There is no solution for 6-digit primes. The chains can also be constructed when  p  and  q  have a different number of digits.  In that case I could find more interesting chains.

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J. K. Andersen wrote:

Q1.
For n = 1000:
10^999+2041297 & 10^999+4511947
11*10^1998+24054111299477 & 11*10^1998+42501411924977
Prp's were found by PrimeForm/GW and proved by Primo.

For n = 2500:
44315126*10^2492-1 & 65581010*10^2492-1
4645351851102060*10^4984-1 & 6454538115010296*10^4984-1
Primes were found and proved by PrimeForm/GW.
The form was chosen to give fast prp tests and easy proofs.

Q2.
All solutions up to n=6:
3 & 7
11 & 23
23 & 47
29 & 71
43 & 97
211 & 283
359 & 509
2129 & 4937
2437 & 3361
19231 & 29017

If the original numbers do not have to be prime then there are
two cases for n=6: 130149 & 475399, and 560131 & 740523

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