Problems & Puzzles: Puzzles

Puzzle 593. Primes by transportation Lost in an old page of my files I found a note, entitled "primes by transportation" 17333 - 1733 3 733 13 73 313 3 3137 - 31337 Q. Can you find a larger prime than 17333 and the corresponding transportation table?

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Contributions came from Anton Vrba, Jan van Delden, Paul Schmidt, Emmanuel Vantieghem.

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Anton wrote:

Three ways of transporting the prime 337397, note the first result is to itself.

337397  -      337397   -     337397   -
33739   7       37397   3      37397   3
 3739   73      3739    37     3739    37
 373    739      739   337     373    937
 37    3739      73   9337      73    9337
  7   33739       3   93377     7    93337
  -   337397      -  393377     -   793337

I like the first one and I ask is this always possible? (CR) 

***

Jan wrote:

I investigated primes where the first or last digit of the left prime is transported to the head or tail of the right prime and all resulting primes different.

So for instance your example would change into:

 

Primes on left side             Primes on right side

 

[17333, 1733, 733, 73, 7, ] [ ,3, 31, 331, 3331, 73331]  or

[17333, 1733, 733, 73, 7, ] [ ,3, 31, 331, 3331, 33317]

 

Solutions starting with the smallest possible prime (on the left side):

 

[37, 3, ] [ ,7, 73]

[37, 7, ] [ ,3, 73]

 

[137, 13, 3, ] [ ,7, 17, 317]

[137, 13, 3, ] [ ,7, 17, 173]

 

[1373, 137, 37, 7, ] [ ,3, 13, 313, 3137]

[1373, 137, 37, 7, ] [ ,3, 31, 331, 7331]

[1373, 137, 37, 7, ] [ ,3, 31, 313, 3137]

 

[13997, 1399, 139, 13, 3, ] [ ,7, 97, 997, 1997, 19973]

 

[317333, 17333, 7333, 733, 73, 7, ] [ ,3, 31, 331, 3331, 33331, 733331]

 

[1996793, 199679, 99679, 9967, 967, 67, 7, ] [ ,3, 13, 139, 1399, 13999, 613999, 7613999]

[1996793, 199679, 99679, 9679, 967, 67, 7, ] [ ,3, 13, 139, 1399, 13999, 613999, 7613999]

 

[33697397, 3369739, 369739, 69739, 9739, 739, 73, 3, ] [ ,7, 37, 337, 6337, 96337, 963379, 7963379, 37963379]

[33697397, 3369739, 369739, 69739, 9739, 739, 73, 3, ] [ ,7, 37, 337, 6337, 96337, 963379, 7963379, 79633793]

No solutions for 9 digits.

***

Paul wrote:

9767699        -
967699         7
96769         97
9679         967
967         9697
97         96697
7         966997
-        9667997

***

Emmanuel wrote:

3293632636633636393366993663663636963933633636363, -
329363263663363639366993663663636963933633636363,  3
39363263663363639366993663663636963933633636363,  23
3936326366336639366993663663636963933633636363,  233
393632636633663936699366366363693933633636363,  2633
39363263663366393669936636636363933633636363,  26393
3936326366336639366936636636363933633636363,  269393
336326366336639366936636636363933633636363,  2699393
33632636633639366936636636363933633636363,  26996393
3363263663363936693636636363933633636363,  269963693
336326366336393669363663636393363336363,  2669963693
33632636633639669363663636393363336363,  26699363693
3363263663369669363663636393363336363,  266993636933
336323663369669363663636393363336363,  2669963636933
33632366339669363663636393363336363,  26699636369633
3363266339669363663636393363336363,  266996363696333
336266339669363663636393363336363,  2663996363696333
36266339669363663636393363336363,  32663996363696333
3626633966936366363639363336363,  326639963636936333
362663396693636636363936336363,  3266399363636936333
36266339669363663636393636363,  32663993636363936333
3626633966936366363693636363,  326633993636363936333
362663396693636636369363363,  3626633993636363936333
36266339693636636369363363,  36266339693636363936333
3626633969363663639363363,  362663396936366363936333
362663396936366363936333,  3626633969363663639363363
36266339693636363936333,  36266339693636636369363363
3626633993636363936333,  362663396693636636369363363
326633993636363936333,  3626633966936366363693636363
32663993636363936333,  36266339669363663636393636363
3266399363636936333,  362663396693636636363936336363
326639963636936333,  3626633966936366363639363336363
32663996363696333,  36266339669363663636393363336363
2663996363696333,  336266339669363663636393363336363
266996363696333,  3363266339669363663636393363336363
26699636369633,  33632663396693636363636393363336363
2669963636933,  336326633966936636363636393363336363
266993636933,  3363266339669366363636363933633636363
26699363693,  33632663396693366363636363933633636363
2669963693,  336326633966933663636363639333633636363
269963693,  3363266339666933663636363639333633636363
26996393,  33632663396669336636363636639333633636363
2699393,  336326663396669336636363636639333633636363
269393,  3363266633966693366363636366393336336369363
26393,  93363266633966693366363636366393336336369363
2633,  933632666339666933663636363663933363369369363
233,  9336326663396669336636636363663933363369369363
23,  93363266633966693366336636363663933363369369363
-, 9336326663396669336633663636366393336336329369363

One day after he added:

CR asks if there are other cases in which the first prime equals the last one.  The answer is yes.  Here I give one such case (I can give 26 other ones) :

 

3293632636633636393366993663663636963933633636363 -

329363263663363639366993663663636963933633636363  3

39363263663363639366993663663636963933633636363  23

3936326366336639366993663663636963933633636363  233

393632636633663936699366366363693933633636363  2633

39363263663366393669936636636363933633636363  26393

3936326366336639366936636636363933633636363  269393

336326366336639366936636636363933633636363  2699393

33632636633639366936636636363933633636363  26996393

3363263663363936693636636363933633636363  269963693

336326366336393669363663636393363336363  2669963693

33632636633639669363663636393363336363  26699363693

3363263663369669363663636393363336363  266993636933

336323663369669363663636393363336363  2669963636933

33632366339669363663636393363336363  26699636369633

3363266339669363663636393363336363  266996363696333

336266339669363663636393363336363  2663996363696333

36266339669363663636393363336363  32663996363696333

3626633966936366363639363336363  326639963636936333

362663396693636636363936336363  3266399363636936333

36266339669363663636393636363  32663993636363936333

3626633966936366363693636363  326633993636363936333

362663396693636636369363363  3626633993636363936333

36266339693636636369363363  36266339693636363936333

3626633969363663639363363  362663396936366363936333

362663396936366363936333  3626633969363663639363363

36266339693636363936333  36266339693636636369363363

3626633993636363936333  362663396693636636369363363

326633993636363936333  3626633966936366363693636363

32663993636363936333  36266339669363663636393636363

3266399363636936333  362663396693636636363936336363

326639963636936333  3626633966936366363639363336363

32663996363696333  36266339669363663636393363336363

2663996363696333  336266339669363663636393363336363

266996363696333  3363266339669363663636393363336363

26699636369633  33632366339669363663636393363336363

2669963636933  336323663369669363663636393363336363

266993636933  3363263663369669363663636393363336363

26699363693  33632636633639669363663636393363336363

2669963693  336326366336393669363663636393363336363

269963693  3363263663363936693636636363933633636363

26996393  33632636633639366936636636363933633636363

2699393  336326366336639366936636636363933633636363

269393  3936326366336639366936636636363933633636363

26393  39363263663366393669936636636363933633636363

2633  393632636633663936699366366363693933633636363

233  3936326366336639366993663663636963933633636363

23  39363263663363639366993663663636963933633636363

3  329363263663363639366993663663636963933633636363

- 3293632636633636393366993663663636963933633636363

 

It took only a few minutes to find this. Note that all the primes in both columns are the same ! 

He also sent a remarkable one in which the end prime is the reverse of the original:

     39366937373  -

     3373396769   3

     337339669   37

     33739669   373

     3373969   3673

     337369   36793

     33769   336793

     3769   3363793

     379   33663793

     37   336693793

     7   3336693793

     -  37373966393

 

 

 

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