Problems & Puzzles: Puzzles

Puzzle 134.  The 1379-Carrousel-Primes

Let's ask for primes P that remains prime when the following set of transformations applies at the same time:

a) all the "1" digits transforms in "3" digits
b) all the "3" digits transforms in "7" digits
c) all the "7" digits transforms in "9" digits
d) all the "9" digits transforms in "1" digits
e) all the other digits (0, 2, 4, 5, 6 & 8), remain the same ('inactive' digits)

Let's ask only for these primes P that remain prime under 3 successive applications of the complete before mentioned set of transformations.

Here are some easy examples:

I. The least carrousel prime with these properties (and at least on 'active' digit) is: 19 -> 31 -> 73 ->97. Incidentally this carrousel prime has all its digits 'active' (to be honest the least carrousel primes are 2 & 5)

II. The least carrousel palprime is: 131 -> 373 -> 797 -> 919. Incidentally this palprime has also all its digits 'active'.

III. The least carrousel prime that generates two couples of reversible primes is:157 -> 359 -> 751 -> 953 (unfortunately it has one 'inactive' digit)

IV. The least carrousel prime with only one digit 'active' is:
821 -> 823 -> 827 -> 829

V. The least carrousel prime with only one 'inactive' digits is: 2111->2333->2777->2999

Questions:

1. Find 3 more carrousel palprimes, having only 'active' digits.

2. Find 3 carrousel primes that generates, each, two couples of reversible primes, having only 'active' digits.

3. Find one Titanic (*) carrousel prime having only one 'active' digit.

4. Find one 100 digits carrousel prime having only one 'inactive' digit.

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(*) As a fair & instructive beginning I can accept smaller solutions if they are composed of 100, 200, ..., etc. digits. As maybe you have noticed these asked four primes may form two couples of twins in the same decade.

Solution

Michael Bell has produced solutions to questions 1 & 3. Here is his email (22/4/01):

"I've found 2 more carrousel palprimes with only active digits:

3193391713171933913 -> 7317713937393177137 -> 9739937179717399379 -> 1971179391939711791 and

7319171773771719137 -> 9731393997993931379 -> 1973717119117173791 -> 3197939331339397913

Also, as a first try at question 3 I found the quadruplet 46060600066600606006*10^30 + 1,3,7,9. Which has 50 digits and only 1 active digit."

Michael is not sure that the palprime gotten is the earliest. He also think that to get a titanic solution to question 3 is "close to impossible, unless someone can see a way of sieving efficiently "

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Giuliano Daddario has solved (2/9/01) the question 2, founding 5 examples with the asked properties:

{139397, 371719, 793931, 917173}

{193937, 317179, 739391, 971713}

{1173193939379177,3397317171791399,7719739393913711,9931971717137933}

{113799319379331977, 337911731791773199, 779133973913997311,

991377197137119733}

{139717391739171397, 371939713971393719,

793171937193717931, 917393179317939173}

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Carlos Rivera wrote on July 13, 2018:

Q1. A Palprime Carrousel Quadruplet, 91 digits all active:
1377373191113371973939373177717339991717199719179917171999337177713739393791733111913737731
3799797313337793197171797399939771113939311931391139393111779399937971717913977333137979973
7911919737779917319393919711171993337171733173713371717333991711179193939137199777379191197
9133131979991139731717131933393117779393977397937793939777113933391317171379311999791313319

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Carlos Rivera wrote on July 20, 2018:

Q1. A Palprime Carrousel Quadruplet, 97 digits all active:

1997117331197931731939739137111917991979131311333331131319791997191117319379391371397911337117991
3119339773319173973171971379333139113191373733777773373731913119313339731791713793719133779339113
7331771997731397197393193791777371337313797977999997797973137331737771973913937917931377991771337
9773993119973719319717317913999793779737919199111119919197379773979993197137179139173799113993779

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Emmanuel Vantieghem wrote on July 21, 2018:

Your beautiful results about puzzle 134 inspired me to look for palprimes that use only two active digits and that produce a caroussel.

This is my best result (so far) :

Q1. A Palprime Carrousel Quadruplet, 101 digits all active of which only two are used :
11333331313131131331313333131131331333311111131311111313111111333313313113133331313313113131313333311
33777773737373373773737777373373773777733333373733333737333333777737737337377773737737337373737777733
77999997979797797997979999797797997999977777797977777979777777999979979779799997979979779797979999977
99111119191919919119191111919919119111199999919199999191999999111191191991911119191191991919191111199

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On June 9 2022, Paul Cleary wrote:

Here are a few more additions to Q1, all of them have a Palindromic Prime number of digits, the first set are all active digits followed by those with only 2 active digits.

101 digits

31797931739317117379193177913937191791371379311973137911397317319719173931977139197371171393713979713

73919173971739339791317399137179313913793791733197379133719739731931397173199371319793393717937191937

97131397193971771913739711379391737137917913977319791377931971973173719397311793731917717939179313179

19373719317193993137971933791713979379139137199731913799173193197397931719733917973139939171391737391

131 digits

19737931173371913791971317713931179317193391717939113971713919733333791931717931193971719339171397113931771317919731917337113973791

31979173397793137913193739937173391739317713939171337193937131977777913173939173317193931771393719337173993739131973139779337197913

73191397719917379137317971179397713971739937171393779317179373199999137397171397739317173993717931779397117971373197371991779319137

97313719931139791379739193391719937193971179393717991739391797311111379719393719971739397117939173991719339193797319793113991731379

151 digits

97131979311937713937139713391793971797317191339317137971937373191913733797111797337319191373739179731713933191713797179397193317931
73931773911397913179193731917331799371793719377139171939197393137717393791931797973131379779193339197797313137979713919739371773139
37919391719317739173971739971337191373913179731397739117939179317993713931713197173799397179131739191973737919913177713199197373791
91937131971793997371791317139317399713971939711937793137971373919737199713391713917391179371739373193979117193913739713131979791311
37399937311319797913131793731939171197939137393717397119371931719331799173791937

and 2 with 181 digits

7991113799917731397373971793313139799313737139791919739119179937371119331113177799733317931397133379977713111339111737399719119379
1919793173731399793131339717937379313771999731119979113337911139973719797193917737371911737979371913131971331391179793331773337399
9119777391737193777911999373337713339797119313317913131917397973711917373771939179791737993111973331191337779133371197931919317139
9797931339791917931373731937737133919177739977797111331999713979317999133111797779937771919331737739137373139719197933139797993171
3919139791173331977733137799913777933191731317393711919173771913139173797973179979377131399971199919333773111937191739111377333919
99117999313177397997137979737193131917737191911739371313719133977731999773

7791113311317997137391317737717919397993131733917731391719933313717913131739333173131797377737971313713339371313197173133399171931
3771933713139979391971773771319373179971311331119779913337733739119379713739979939131719117373977139973713931177737939137373971777
3973739197999791937379377717937373193973777113931737993177937371191713193997993731797391193733773331991137779977971331791937971191
1713739313397971993711979371733999791713797971939997197971319111913179791799939179797317197999337173979117399179793313937317119117
9739197133179779977731133799911991937739131791933133937971737719193117933191793977111913937919193171119319193731333137391913911171
39191973931911177939719133971139191773717973933133919713193773919911999733

2 active digits and 101 digit length.

99999199119919991999199999111911991999199919111999999911191999199919911911199999199919991991199199999

11111311331131113111311111333133113111311131333111111133313111311131133133311111311131113113311311111

33333733773373337333733333777377337333733373777333333377737333733373377377733333733373337337733733333

77777977997797779777977777999799779777977797999777777799979777977797799799977777977797779779977977777

2 active digits and 131 digit length.

19111191191119199111111119991999991919919919191991191191111911999199911911119119119919191991991919999919991111111199191119119111191

31333313313331311333333331113111113131131131313113313313333133111311133133331331331131313113113131111131113333333311313331331333313

73777737737773733777777773337333337373373373737337737737777377333733377377773773773373737337337373333373337777777733737773773777737

97999979979997977999999997779777779797797797979779979979999799777977799799997997997797979779779797777797779999999977979997997999979

2 active digits and 151 digit length

9991919119199919199999191911191111119191919919991191199119911111999991919119119191999991111199119911911999199191919111111911191919
9999191999191191919991113131331311131311111313133313333331313131131113313311331133333111113131331331313111113333311331133133111311
3131313333331333131311111313111313313131113337373773733373733333737377737777773737373373337737733773377777333337373773773737333337
7777337733773773337337373737777773777373733333737333737737373337779797997977797977777979799979999997979797797779979977997799999777
779797997997979777779999977997799799777977979797999999799979797777797977797997979777

2 active digits and  181 digit length

999991919191111919119199999919911111199111111991999119199991911199911991991999111111919991119991911111199919919911999111919999191
199919911111199111111991999999191191911119191919999911111313131333313133131111113113333331133333311311133131111313331113311311311
133333313111333111313333331113113113311133313111131331113113333331133333311311111131331313333131313111113333373737377773737737333
333733777777337777773373337737333373777333773373373337777773733377733373777777333733733773337773733337377333733777777337777773373
333337377373777737373733333777779797979999797997977777797799999977999999779777997977779799977799779779777999999797779997779799999
9777977977997779997977779799777977999999779999997797777779799797999979797977777

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