Problems & Puzzles: Puzzles

Puzzle 1045. One nice puzzle from Paolo Lava

Paolo lava sent the following very nice puzzle.
 

Let p a zeroless prime with k digits (d_1, d_2,Ö,d_k). For any digit, delete it from p and divide* the rest by the digit itself. Take the sum of these numbers and check if it is a prime. E.g.

p = 21673 -> 1673/2 + 2673/1 + 2173/6 + 2163/7 + 2167/3 = 4903 that is prime.

The first primes of this kind are 11, 21673, 27367, 32611, 33311, 41141, 48821, 82781, 171263, 211441, 243433, 323443, 343243, 449699, Ö

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*real division

Q1. Can you extend that list?

Q. Are there chains of 3 or more primes we can get reiterating this process ?


During the week from 11 to 16 July 2021, contributions came from Giorgios  Kalogeropoulos, Adam Stinchcombe, Simon Cavegn and Paolo Lava, Jan van Delden and Emmanuel Vantieghem.

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Giorgios wrote:

Q1. This sequence is now A346206.
       If we search the first 10^9 primes we will find that 28997 of them belong in this sequence.
       I am sending you those 28997 first terms in a txt file.
       I don't know if this is an infinite sequence but we can find very large terms like this 100-digits term: 
       1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111117678381

 
Q2. I found 3 chains of 3 primes: 
       {36388174733, 12847141847, 7987772867}
       {237447416471, 113199661229, 71718711151}
       {241291644949, 116646463163, 68281231591}      

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Adam wrote:

Starting at half a million and going up to  2592741971 I found 5777 such primes, starting with  632623, 663661, 727271, 772127, 847871, 882881, 944969, 1129699, 1192699, 1193939   and ending with   2499986821, 2525355533, 2525562553, 2553525353, 2555253353, 2555525333, 2555532533, 2555553323, 2555656651, 2556255523.  I did not find any three step primes.

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Simon wrote:

Q1 Found over 9 million solutions.

Q: Found these chains:
36388174733, 12847141847, 7987772867
66842966899, 13416316643, 8806041977
168626231383, 132121131223, 124205334317
232739362799, 81255554851, 31788470489
237447416471, 113199661229, 71718711151
241291644949, 116646463163, 68281231591
379217996627, 143894291389, 90704472083
398396183491, 178679982947, 113234467039
464432264113, 236372466431, 105172127689
466618184741, 221617784813, 117051517259
693919699393, 211831418621, 144895922783
1187174316871, 899883194381, 338668447567
1843994199989, 1346137772461, 946452821689
2228134782767, 994664339233, 297380927509
2232139966619, 1124628881483, 643999693219
2248128271673, 1212373227661, 849982242817
2284848163811, 1312662313663, 964385019779
2334399164291, 1218328146263, 832173191459
2419819898383, 1119621924419, 806670615521
2826617482267, 1478348641727, 992857601851
2832849488993, 1118466462481, 690896699573
2889189983131, 1744788332347, 1251676538941
2893911968641, 1749338991683, 1296513408581
3988443941191, 2189499383263, 931654791883
4322981398693, 1699438628119, 1384017846433
4371777472373, 1681268343221, 1523397911027
4447362727141, 2193839414189, 1212980863027
4466327366663, 1413448893799, 852842365493
4554952955933, 1391242934639, 964365218891
4761774488123, 2177989924867, 787079448781
8146819723699, 3324323331463, 1657671410581
8274682723217, 3618668991493, 1532646704299
8361336843443, 3292991998189, 1418910394513
8464384348681, 2918923148293, 1771896703697
8691239967181, 4233199982821, 2074009948471
8831497696729, 2888673777683, 1042633114097
9333386843939, 2733418741843, 1532523549961
9416326429921, 4648111843211, 3297459949747
9732318819497, 4138122628441, 2439622846499
9917764988333, 3248232278237, 1379056409483
9957556525229, 3121361121611, 2754703393349
9999236349149, 3499434481849, 1286890187659
11294267492297, 6331336364473, 2571688991899
11666269198489, 6832261813241, 4117539887099
11681233631383, 8912492641489, 4005901734059
11932123393319, 9823124983921, 4886834136661
12116862862463, 8138261949619, 4010547173533
12268849629139, 7218673884841, 3027018896393
12461223769499, 7734343284827, 2640888933209
12966284829139, 8484168664841, 3356533387117
13342298869139, 8684183848841, 3452748896917
13473912191419, 11799216769363, 7477921497619
13612464929639, 9134392228891, 4501911164063
13823874217633, 9889143361973, 4085845549699
14116732882117, 12926496242923, 8408576644123
14246193269639, 9911246782891, 5079717793151
14294428296289, 9295655562539, 2750889123343
14399739326747, 8626613642821, 4279755502471
14484494969863, 8123169481969, 3986723865653
14611119994111, 16329217964149, 13861862826079
16281696491689, 12763114788211, 11026568848843
16333293694889, 11183271827467, 7383451935247
16382268939649, 11268268288261, 7064755982191
16441212993629, 14626636868231, 10514259180827
16621882343441, 14919927966499, 9715948142603
16848999616733, 12168299148911, 9496695497687
16899232264819, 13694824329983, 8257874755309
17262762687187, 13811286413233, 12233058014741
17366828434643, 12782231172383, 10109944273123
17479239797431, 13691392722913, 10843217165243
18392191921381, 19848369239213, 17768831107361
18396284699263, 13878236427733, 8798754294749
18429627899171, 16276331372627, 13449982613843
18832436886463, 14469293314493, 10268547275167
18871477111127, 21268342341383, 11962505210231
18968946984283, 13717428863323, 9709437053501
19169243139143, 19681889877179, 16400352585109
19626391983383, 16944739739173, 12790679611777
19822232229389, 18221418463747, 16866132268369
19822647899111, 19992346299439, 15987814200607
21126627876787, 9722938912483, 3993958453801
21216231672113, 16919838886129, 13725569087777
21899842677191, 12141614189989, 9114032382803
22229269396613, 11127268148131, 9106625474473
22616618328461, 13922368669981, 9312347070053
22661464961929, 11724813374387, 7757115981679
22722273233111, 16997776936123, 12874807809167
22744712767237, 11188611843481, 8600641614307
22828136491969, 11955156995653, 7492274911801
22897129791383, 12138334138261, 9409523979827
22944432912439, 12116641761421, 10733822347327
22982964891631, 12174111412447, 11078326048289
22988493314491, 12182661286463, 8320080212419
23131188842483, 14244671448823, 10111290924187
23988869836933, 8187194336939, 3381180584059
24863183712733, 14421223244663, 11621037611179
24877132474687, 11419141771699, 8875591879717
24888369267989, 8499334683149, 2943160396699
25455472155721, 14894439473987, 8772560598289
26336664446881, 12338662117781, 9224561245243
26417191819819, 19398216146891, 19056358363429
27726327213671, 17912938927483, 14704900684549
28672378666643, 12299384136839, 7288475003561
28886833441663, 14553133554653, 10397930504081
 

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Paolo wrote:

it's easy to extend the sequence. Here below the primes P< 10^6 with their produced primes Q. I noted that digit 5 never appears in P....Just for your info: the puzzle is now in A346206, uploaded by Michel Marcus (see also A346217)...

I was doing some tests and I found 515.571.557, 551.571.551, 555.757.511. For sure  515.571.557 is not the least prime with digit 5 because I did not perform an exhaustive test.
Other greater values of primes with digit 5 are 5555665133, 5556163553, 55556192959,..., 55556226533, 55556295199, 55556381581, 55556564641, 55556788453, 55556859989, 55556884351, 55558486351, 55559519339, 55559651929, 55559689589, 55559913593, 55559929651, 55559951393, 55559965921, 55559995373, ..., 555556113163, 555557141147, 555558417877, 555559223993, 555559661399, 555559923329, 555561185783, 555561356131, 555561944599, 555563195699, 555563516713, 555563737657, ... Note that all these primes have exactly five digit 5...the least prime with digit 5 is 155.455.541 that produces 94.285.909.

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Jan wrote:

One could reduce the search space if one uses the following necessary condition:
 
The number of digits equal to 5 of a number (not ending in 0 or 5) must be 0 modulo 5 in order to have a successor.
 
A rule for the divisor 2 (of 10) can also be deduced, but I didnít use that (it is not that nice).
 
I found the following chains of length 2:
 
36388174733  12847141847   7987772867
237447416471 113199661229  71718711151
241291644949 116646463163  68281231591
693919699393 211831418621 144895922783
 
The number of primes that generate a prime for which both primes fulfill this condition; a necessary condition to find a sequence of length 2:
 
2:      1
6:      4
7:     32
8:     83
9:    450
10:  2273
11:  9735
12: 50971
 

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Emmanuel wrote:

I listed  2544  primes < 10^9  that are mapped on a prime by paolo's procedure (in annex).
I could not find any prime < 6*10^9  that maps on a prime that in turn maps on a prime.
In my opinion this is quite remarkable.  Could there be some theorem behind ?
 

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