Problems & Puzzles: Puzzles

Puzzle 1172  Follow up to Puzzle 1171 Paolo Lava sent the following nice puzzle as a follow up to Puzzle 1171 Let p a prime with k digits (d_1, d_2,…,d_k). For any digit, delete it from p. Take the sum of these numbers and check if it is a prime.    Q. Are there chains of 8 or more primes we can get reiterating this process ?   First chains with 7 primes for  17577607,  22977607, 36992723, 65794423, 71564809, 90272807, ...  e.g.   17577607 -> 7577607+1577607+1777607+1757607+1757607+1757707+1757767+1757760 = 19721269 that is prime; 19721269 > 9721269+1721269+1921269+1971269+1972269+1972169+1972129+1972126 = 23223769 that is prime; 23223769 -> 3223769+2223769+2323769+2323769+2322769+2322369+2322379+2322376 = 19384969 that is prime; 19384969 -> 9384969+1384969+1984969+1934969+1938969+1938469+1938499+1938496 = 22444309 that is prime; 22444309 > 2444309+2444309+2244309+2244309+2244309+2244409+2244439+2244430 = 18354823 that is prime; 18354823 > 8354823+1354823+1854823+1834823+1835823+1835423+1835483+1835482 = 20741503 that is prime; 20741503 > 0741503+2741503+2041503+2071503+2074503+2074103+2074153+2074150 = 15892921 that is not prime;

---

From April 20 to 26, contributions came from Michael Branicky, Simon Cavegn, J-M Rebert, V. F. Izquierdo, Oscar Volpatti

***

Michael wrote:

I found no chains of 8 for numbers up to 4*10^11

***

Simon wrote:

Chains of 9 primes:
14844921801023
20226510439523
21537965495143
22918913556023
23822605556069
24233848199069
25442041982081
28207217282081
28506292946023
31172673712969
33745661247569
34154274355043
34356967099069
34398250502969
38713782142523
40531440683843
40567784406469
42371945001469
42378036382043
44043485308843
44971629081523
45684588477523
46799843245949
49511965240949
50099554803581
50461946018023
50651914482469
50794618598081
51415046679023
51570630477523
52513178392043
54350316150349
55593443912869
56568010484683


Chains of 8 primes:
10727046812081
11284486879469
11392744499569
11852666446069
12272852359069
12298632911069
12826502617081
13525424214569
13927997499569
14108969812081
14338970780143
14368355214569
14549176806023
15547887013543
16040134354523
16338922622969
16387269846643
16571025529469
16955193729469
17022821698543
17196503578643
17236886709569
17389397599069
17475513815581
17660284409569
17669087433643
18197318641643
18340728421069
18441377609069
18636194389523
18650583318949
19050445539523
19080227415043
19237665795143
19659851450143
19683540990143
19739453515043
20325491641643
20327522413543
20388922622969
20426097410581
20781539990143
20802821698543
21047160450143
21166852967081
22390728421069
22391694809569
22572884377969
23455087292569
23626455741023
23645035798949
23821011930449
23899867610581
23995756595143
24091011930449
24313960530449
24389113574069
24464968410581
24649135913569
24678408519469
24753123149143
24921347038949
24964584493543
24999902104069
25066251352081
25144608125069
25189480650449
25250644997969
25289130898949
25377235194469
25488957668569
25540114068569
25567304134523
25798462476023
26165717239069
26231671889569
26350354921069
26576477440043
26632675502969
26688288170069
26732370780581
26735343089569
26755137639143
26770028399069
26911753499069
26988625281643
27038987585069
27359034387569
27383720882081
27425589645043
27733590632081
27889743355043
27997674799069
28010382246023
28021713314069
28205103758569
28252687995449
28305015089569
28325284719469
28352156185069
28414847168569
28537379825069
28639567550449
28741202287081
28772208354469
29106055746023
29143192550449
29453641555043
29532762925069
29622683977343
29724692852843
29737590444143
29909964299143
30029517392969
30136155113569
30624160982081
30674656520143
30752079729523
30804413685449
30818987585069
30952973639069
30995338480069
31000119697643
31059431568569
31082777289569
31083253354469
31091016185069
31146680136023
31337629341023
31346696520143
31779623040043
31953807994469
31997349902569
32060382246023
32218156920143
32219225782643
32246000194469
32255103758569
32304235550449
32356367234143
32499917529523
32641149095449
32685427849069
32882295695581
32992254588143
33193192550449
33217893920143
33274514582081
33436477439143
33498685697969
33763776773449
33807234897523
33868901946023
33990344298949
34041076519523
34042797838949
34090319792569
34121791714069
34160759778143
34338749398949
34495171380581
34541579792969
34553921038543
34561847677523
34703497285969
34869288047569
34951743899069
35056687188143
35126552885449
35129007276023
35173322782643
35559670589543
35575054644769
35716300182643
35782352931023
35834496113569
35991432817969
36110383498949
36222719098949
36225094847569
36314814697643
36402580158569
36496161978449
36496184127523
36511310642449
36548598033769
36550601404081
36808612875523
36844546545949
36960401455949
36980249458969
37007308284769
37026874534543
37052973262069
37068177340343
37083594818843
37165139799769
37211251399769
37216377708449
37217508579023
37221336330469
37228975272023
37310639858969
37326162477523
37352351655469
37358757140543
37476794547449
37615036928023
37675362739081
37738738018769
37743645678449
37844332167469
37889684134949
37898741399023
37898741408581
37905954433769
37924894086043
37952909315543
37958162635343
37994340156343
38006988207449
38056318308581
38129124049769
38249275287043
38250981687043
38252339050769
38265432534949
38343147597523
38356490778449
38410084023581
38491514864969
38498198744081
38501705635343
38523334144081
38554244018023
38571823543843
38617712680769
38634308551043
38735500276769
38905810630523
39001734282023
39029807499769
39044563294769
39109428499769
39175710085969
39240086082023
39341954542343
39352324804081
39364341545569
39494242560469
39533859356543
39591622723843
39594014810569
39639429730543
39643846154543
39647070111469
39653248562069
39700822851143
39756787711469
39844283977343
39885528399949
39911862818843
40143476228543
40179961050043
40223472382081
40270380968569
40543097679023
40561310642449
40591165399969
40608015728023
40663113336581
40799455841581
40861778983843
40876578118969
41011451993081
41138724687581
41140577950949
41214221228449
41286451228843
41300082488081
41303107477523
41315060945843
41396898882023
41424014962069
41543094366343
41567009308843
41622720773449
41637113346581
41665036928023
41677924223449
41763594818843
41782848323449
41801498233843
41803777875523
41804655485969
41809348815469
41824530882469
41871319144769
41993324049769
42005155135523
42141991682023
42266936886043
42314606766343
42330765177523
42337557603269
42371527739081
42389473347343
42426270745949
42551705635343
42565777844081
42574534829843
42618342989543
42646478778269
42814046465543
42862830894769
42863859156043
42934570284643
42980886488081
43039768201769
43102358742343
43111360208269
43238429675843
43318059139343
43325450950949
43337536900081
43363984628843
43374628150949
43382072197523
43500600550543
43500810139343
43530610018843
43543573006469
43613247030569
43614599125081
43658863723843
43697070111469
43704064125343
43714681684643
43726770045949
43827608956343
43868497146581
43961862818843
43997078987843
44071880261449
44095474788769
44097608956343
44128770228449
44145262897523
44240023741523
44248072623023
44255582673581
44294411401469
44363999670469
44418914282023
44487137644769
44492276702581
44541829377581
44569292934949
44591570430523
44603839550269
44626976224843
44630298218843
44699439481043
44791366654543
44904207330469
44926802401469
44928241030469
45071951556343
45151225799969
45181186288081
45322030294949
45361611516469
45454621245949
45488057082023
45502377889769
45548742118969
45609645406543
45628928686043
45688298145949
45716606077523
45831581626949
45864687342343
45895151866469
45919972587043
45985927302269
46064067728843
46097341845769
46104453630523
46174248556043
46192001187043
46193056149769
46218800945843
46396741628449
46473407346449
46507084426949
46525104627523
46662757310843
46733302289969
46736916823043
46779480761449
46803980752843
46829180655469
46838758679023
46878774720343
46953201441581
46968452044769
46990591100081
47017338601769
47018396134769
47020107051143
47068812934769
47096874076523
47172071465969
47231447799023
47240103702581
47254313401469
47274320123843
47282633900081
47303394302581
47328581361449
47405019556043
47405980645843
47431628630569
47502695798081
47503905499769
47541445589543
47627995915543
47648848150949
47744156426423
47769658575481
47800444104323
47800939132123
47931150889543
47949038944769
47958391754543
48033324468383
48066842114623
48087712386943
48175320140543
48207500062343
48232498897523
48277707142523
48331262472023
48355566794023
48363438261269
48379466592043
48497523418769
48564508206343
48586527225343
48635548172023
48737687344081
48744220654543
48782721829843
48798320254969
48841859462069
48883979916469
48906172116469
48935129001469
48959062497523
48965076139543
49019703583769
49050179144081
49052687640949
49059911156143
49109039552843
49130699281523
49161927420343
49216592823581
49277081528023
49284760887023
49321590102581
49400656525081
49400999433769
49435118395469
49528752728843
49592763761269
49596049472023
49605897845843
49761539971123
49797904962749
49874560238981
49935066187183
49965881040349
49983251498083
50009962741523
50066530262069
50248845040769
50250787462069
50303443583843
50322397887469
50332822270469
50335704871769
50339758662343
50379916153969
50397825803449
50445893556143
50557084426949
50560184102581
50630954730469
50646596827523
50709807556543
50735991835343
50772862400081
50783302289969
50888758679023
50951410206343
51070107051143
51099746673581
51145421098081
51155249799949
51218279254543
51224065604081
51232670736581
51300593626949
51348886162069
51372953749769
51378765569843
51407204303449
51469324239643
51486919669543
51558278244769
51589723679023
51615982792043
51666320823449
51710249035343
51778972235381
51840854948483
51850444104323
51950808242449
51977657250269
52037221641343
52128123614423
52221284833769
52276587055949
52298176680769
52312030525081
52353374171543
52353604429843
52360291356143
52401698695469
52403838206543
52551394240769
52585777740343
52760548956043
52830179144081
52832687640949
52865779679081
52968276603449
53065729080769
53099268903269
53111938628843
53119617350543
53133714076523
53230058635343
53311544054969
53323821002449
53330445630523
53337038554543
53352171235343
53355736679081
53413791628843
53423514199969
53473659039643
53476281528023
53492756539543
53497964539343
53534899118969
53578752728843
53595324350543
53677493892383
53828929867123
53858751430823
53859932687749
53881938757349
53884388537549
53917628178181
53975534900381
53977658905249
53979928182923
53985066187183
54007424419181
54018123614423
54018826904323
54066680693383
54118423291369
54160000668383
54182020881883
54204376667923
54251387685083
54319144587881
54353050767923
54391152776881
54436558238981
54437834187923
54490440694883
54510172452349
54553014714323
54628627091183
54637983670883
54671331465349
54745377873323
54784148078981
54799117550483
54878228527823
55073040473443
55088401937549
55103584842749
55105297180883
55117619595349
55156356182881
55186231505849
55252741779923
55281473832383
55362761890481
55424428069181
55458289613983
55468006492123
55480772415623
55501534292123
55518658633981
55595124423583
55605117477383
55741024253483
55752003724423
55754449489381
55827586372823
55835018821823
55884921677383
55922516033981
56048337137323
56067196533983
56162359938583
56162677521823
56174980161823
56200869120623
56228342952869
56239116292123
56239116301681
56421151673323
56442761426423
56500953295481
56524744448981
56599337187881
56600673413849
56646680614423
56669499479249
56670830075623
56690873179883
56712915979883
56759523159923
56776381409323
56966507590549
57068145893383
57077695589683
57078334811123
57094244673443
57109166290481
57117206830483
57232240948483
57256041628381
57279055859923
57322005876983
57354039898049
57370954592881
57393610796749
57403625711983
57502396161823
57507019372823
57563107214869
57620426015623
57649449923123
57726513626423
57788596587923
57807996632549
57824771629423
57856478725681
57859061582923
57930057450481
57957453931123
57981595340083
58066983914623
58094771629423
58109314182923
58127396589683
58168296782881
58293908516183
58353558725681
58367566682881
58451911994869
58550298961823
58625662047383
58626045340381
58651637955349
58706147622823
58708232452481
58714153583881
58717388174381
58766598349369
58768824174923
58807140014423
58829312452481
58831313682481
58832407296749
58845698282323
58846210782923
58877351466823
58889535638123
58897978671943
58956845438983
58959930165683
58961524583749
59043732867923
59107962479249
59123346173981
59129335384181
59134863332123
59167544023123
59170578801623
59228053838981
59244738030823
59258573865083
59270606393981
59327891156623
59408765579923
59427416982881
59538913495249
59544991678981
59570611050481
59587426973083
59640708515981
59651004913849
59768390987183
59828583168181
59897936156623
59999707126823
60022121675083
60034950292123
60038098894883
60063720078443
60158279516183
60159709526183
60203148833981
60221433885883
60230174777923
60234910795883
60241973141681
60242526359623
60254495141681
60262991879249
60327361750349
60346627178683
60346767642383
60347332062923
60349913421823
60386616289249
60388258492123
60492406656623
60594806440249
60599195076983
60611757923881
60614832075883
60634843882183
60681821500849
60695684711983
60742998036883
60746748165349
60750501115981
60851119037383
60857695589683
60977596641883
61152514352749
61177867512869
61297757282323
61317815666383
61372005876983
61453625711983
61485859073323
61554766463849
61586670644249
61612884900823
61653759628849
61667698722823
61699449923123
61761753272383
61762239288983
61783114324423
61805830697749
61827248073443
61841141593181
61845712840349
61860025232549
61894263527881
61907794746983
61917287352749
61948296782881
61997614080683

***

Rebert wrote:

Starting prime of a n chain:

n p

2 11

3 29

4 199

5 39631

6 16296109

7 17577607

9 20226510439523

 

I found the following 9 chain:

 

20226510439523 (14-digit prime)

 -> 26732370780581 (14-digit prime)

 -> 41502015708581 (14-digit prime)

 -> 55759911038981 (14-digit prime)

 -> 78419394975461 (14-digit prime)

 -> 109926481764719 (15-digit prime)

 -> 172628751927997 (15-digit prime)

 -> 310012362449357 (15-digit prime)

 -> 443064323522113 (15-digit prime)

 

***

V. F. wrote:

My only contribution to this puzzle is the following: The 1st sum of digits according to condition, gives the results:

NºDígitos  1ª suma

2         a+b

3         20 a+11 b+2 c

4         3 (100 a+40 b+7 c+d)

5         4000 a+1300 b+220 c+31 d+4 e

6         50000 a+14000 b+2300 c+320 d+41 e+5 f

7         3 (200000 a+50000 b c+1100 d+140 e+17 f+2 g)

8         7000000 a+1600000 b+250000 c+34000 d+4300 e+520 f+61 g+7 h

9         80000000 a+17000000 b+2600000 c+350000 d+44000 e+5300 f+620 g+71 h+8 i

10        9 (100000000 a+20000000 b+3000000 c+400000 d+50000 e+6000 f+700 g+80 h+9 i+j)

11        10000000000 a+1900000000 b+280000000 c+37000000 d+4600000 e+550000 f+64000 g+7300 h+820 i+91 j+10 k

12        110000000000 a+20000000000 b+2900000000 c+380000000 d+47000000 e+5600000 f+650000 g+74000 h+8300 i+920 j+101 k+11 l

13        3 (400000000000 a+70000000000 b+10000000000 c+1300000000 d+160000000 e+19000000 f+2200000 g+250000 h+28000 i+3100 j+340 k+37 l+4 m)}

14        13000000000000 a+2200000000000 b+310000000000 c+40000000000 d+4900000000 e+580000000 f+67000000 g+7600000 h+850000 i+94000 j+10300 k+1120 l+121 m+13 n}

15        140000000000000 a+23000000000000 b+3200000000000 c+410000000000 d+50000000000 e+5900000000 f+680000000 g+77000000 h+8600000 i+950000 j+104000 k+11300 l+1220 m+131 n+14 o

16        3 (500000000000000 a+80000000000000 b+11000000000000 c+1400000000000 d+170000000000 e+20000000000 f+2300000000 g+260000000 h+29000000 i+3200000 j+350000 k+38000 l+4100 m+440 n+47 o+5 p)}

...

It seems that if the number of digits is of the form 3k+1, it cannot be prime.

***

Oscar wrote:

The first chain of length 8 begins with prime 10727046812081:

 

10727046812081, prime

15284620515581, prime

24963218402081, prime

37845991221581, prime

57070332504881, prime

80681960501141, prime

106190526261917, prime

160063608146509, prime

278321850240026, composite

 

Next five chains of length 8 begin with the following primes:

11284486879469,

11392744499569,

11852666446069,

12272852359069,

12298632911069,

 

I was hoping to find chains of infinite length by detecting periodic loops.

I found some loops with small periods, but containing composites too.

 

period 1:

1729404, 1729404;

period 2:

1686420, 1713582, 1686420;

period 3:

13968815, 14292656, 14218067, 13968815;

period 4:

119834438, 123355957, 123102491, 122163727, 119834438; 

and a few more

 

Unfortunately, there is no such loop containing only primes ("prime loop"):

no prime loop begins with a prime below 10^11, by exaustive search;

every sequence beginning with a number above 10^11 is strictly increasing, so it can't contain a loop.

 

***