Problems & Puzzles: Puzzles

Puzzle 1171  Delete power and sum Paolo Lava sent the following nice puzzle (2nd version mistakes free): Let p a prime with k digits (d_1, d_2,…,d_k). For any digit, delete it from p and take the power of the remaining number elevated to the deleted digit. Take the sum of these numbers and check if it is a prime. E.g. p = 1483-> 148^3 + 143^8 + 183^4 + 483^1 = 174859125675638597 that is prime. The first primes of this kind are 11, 23, 29, 41, 61, 173, 197, 211, 233, 317, 397, 479, 673, 691, 773, 811, 937, 1103, 1483, 1723, 1801, 2143, 2341, 2389, 2861, 4007, 4027, 4463, 6269, 6287, 6329, 6581, 7027, 7103, 8101, 8669, 8803, 8929, 10513, 10651, 11059, 11171, 11813, 11933, 12251, 12473, 13151, ... Q. Are there chains of 4 or more primes we can get reiterating this process ?   Chain of 3 prime starts from  4027, 29453, 111323, 940183, 1072793, 2663993, 3452431, 5925097, 10179929, 11581807, 12125587, 12139483, 12582289, 13753301, 16897627, 22676089, 23264587, 24117887, 25751303, 25975087, 30920861, 33347827, 34811261, 34739321, 34840523, ... E.g. 4027 -> 402^7 + 407^2 + 427^0 + 027^4 = 1696611363948396419 that is prime1696611363948396419  > 169661136394839641^9 + 169661136394839649^1 + ... + 696611363948396419^1 = 4483650302940486858662531420954819299488335158733901706168642723148937 86163378579506300993827830318985609668193935755801524813863578002591546206103429995469 that is prime.

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From April 13 to 19, 2024, contributions came from Simon Cavegn, Michael Branicky, VIcente Felipe Izquierdo, Giorgos Kalogeropoulos, J-M Rebert, Emmanuel Vantieghem.

 

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Simon was the first and the only that noticed that the original puzzle had some mistakes:

For the current puzzle 1171, I get different numbers? May be I'm doing something wrong, or the Question has wrong numbers:

 

Test failed for 1201, not prime:10523
Test failed for 1409, not prime:20661046784141158571
Test failed for 4801, not prime:668582463647240326883
Test failed for 5801, not prime:3969189584115133408583
Test failed for 6203, not prime:69980607583939
Test failed for 7307, not prime:110731006202781509687

Test failed for 1103, prime:1331207
Test failed for 1801, prime:10828567056281783
Test failed for 7103, prime:122987744454191
Test failed for 8101, prime:10828567056282413
Test failed for 8803, prime:345743765739145428838723

Later he added:

Found chains of 4 primes, using a probabilistic prime test:
4818883183
17447639119
30643023353
377925860029
393611685983
397756416667
1062422293811
1304097629657
1462569837679
1908240354613
2184294198059
2184809465833
2201764030651
2516950300133
3033361456673
3087604263673
3139407123793
3634847693311
3828119593097
3922121331811
4120251225317

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Michael Branicky wrote:

4818883183 starts the first chain of 4 primes... and he sent an annex with the detailed prime thus obtained.

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Vicente wrote:

Only found for Chain of 4 prime:

   4818883183

17447639119

 

Up to prime[10^9]

Not others chains found >= 4 up to Prime[10^9] (22801763489)

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Giorgos Kalogeropoulos

The first 3 integers that start a 4-chain are:

4818883183, 17447639119, 30643023353
 

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Rebert wrote:
 

I found 6 chains of 4 primes respectively starting at:

4818883183 (10-digit prime),

17447639119 (11-digit prime),

30643023353 (11-digit prime),

377925860029 (12-digit prime),

393611685983 (12-digit prime) and

397756416667 (12-digit prime).

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Emmanuel wrote:

The smallest prime of a chain of four primes is  4818883183.
That prime is mapped on  12578492577020902366821116720014179966261482179585847582603121331860691 (proved prime by Mathematica).

That prime is mapped on a prime of 623 digits.  I think it is worthwhile to get information on its provability. I'm using Mathematica, but after one week of computing I still have no confirmation.

Finally, the fourth "probable prime" of the chain has 5597 digits, Actually, I'm involving PRIMO to get a certification of its primality but the work is also not yet finished.

I expect the (positive ?) result next week.

one day later he added:

A few moments ago PRIMO certified the primality of the 3d and the 4th number of the first 4-primes chain.
 

The verification of the 3d number took 1 minute and 14 seconds.

(there was no result with Mathematica).
 

The verification of the 4th took 164 hours 22 minutes and 15 seconds.

 

 

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