Problems & Puzzles: Puzzles

 Puzzle 1082 Chain of primes such that... Paolo Lava sent the following nice puzzle Take a prime p and convert it to base 2. Then consider any number we can get moving the most significant digit as least significant digit, convert them to base 10 and take their sum. E.g.  p = 53, in base 2 is 110101. Then 101011 -> 43 010111 -> 23 101110 -> 46 011101 -> 29 111010 -> 58 and 43 + 23 + 46 + 29 + 58 = 199. I take into account primes p that generate other primes, like in the previous example: 61 -> 199. The first ten are: 3   3 23   101 53   199 89   419 139   881 163   857 197   823 317   2749 347   2719 359   2707  There are chains of primes: 3 primes: 163 -> 857 -> 5281 4 primes: 317 -> 2749 -> 30011 -> 297659 Q1. Are there chains of this type with 5 or more primes? Q2      Are there two or more primes that generate the same prime? Of course we could perform the same search using different bases. In particular, in base 10 these are the first chains of 3 primes: 181 -> 929 -> 1291 271 -> 839 -> 1381 431 -> 457 -> 1229 433 -> 677 -> 1543 521 -> 367 -> 1409 631 -> 479 -> 1741 Q3. In base 10, can you get primes generating chains of 4, 5, ... prime terms?

During the week 9-15 April, 2022, contributions came from Claudio Meller, Paul Cleary, Giorgos Kalogeropoulos, Oscar Volpatti, Emmanuel Vantieghem.

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Claudio wrote:

Q3. la primera cadena de 4 términos en sistema decimal es:
71411 84143 138077 2750809

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Paul wrote:

Q1.

These are some chains of length 5 and 6.

178403 -> 2443027 -> 39500003 -> 900024079 -> 16279845089

189353 -> 2432077 -> 39510953 -> 900013129 -> 14132372393

581447 -> 12001453 -> 189325127 -> 4105642153 -> 64613834567

631549 -> 11951351 -> 256484089 -> 4038483191 -> 81860862709

649769 -> 9835981 -> 191490599 -> 4103476681 -> 64616000039

698729 -> 9787021 -> 191539559 -> 4103427721 -> 64616048999 -> 1309773485701

742057 -> 9743693 -> 191582887 -> 4103384393 -> 64616092327

944263 -> 9541487 -> 225339523 -> 3532756847 -> 73776654463

2436787 -> 47894849 -> 757411507 -> 16422457661 -> 327174925999

3377587 -> 46954049 -> 758352307 -> 16421516861 -> 327175866799

3722419 -> 46609217 -> 758697139 -> 16421172029 -> 327176211631

3745939 -> 38197091 -> 901326991 -> 16278542177 -> 292959103117

5631037 -> 111809461 -> 1767238717 -> 36887466929 -> 1062624160831

8444719 -> 192881861 -> 4102085419 -> 64617391301 -> 1309772143399

8472047 -> 192854533 -> 4102112747 -> 64617363973 -> 1309772170727

8565743 -> 226315267 -> 3531781103 -> 73777630207 -> 3224757253097

8573743 -> 192752837 -> 4102214443 -> 64617262277 -> 1309772272423

8833003 -> 226048007 -> 3532048363 -> 73777362947 -> 2125245892589

8962799 -> 192363781 -> 4102603499 -> 64616873221 -> 1309772661479

9089863 -> 192236717 -> 4102730563 -> 64616746157 -> 1309772788543

9100079 -> 225780931 -> 3532315439 -> 73777095871 -> 2675001973549

9227143 -> 192099437 -> 4102867843 -> 64616608877 -> 1309772925823

9293227 -> 225587783 -> 3532508587 -> 73776902723 -> 2125246352813

9459661 -> 191866919 -> 4103100361 -> 64616376359 -> 1309773158341

9553007 -> 191773573 -> 4103193707 -> 64616283013 -> 1309773251687

9599017 -> 158173133 -> 4136794147 -> 64582682573 -> 1309806852127

9787021 -> 191539559 -> 4103427721 -> 64616048999 -> 1309773485701

9824327 -> 191502253 -> 4103465027 -> 64616011693 -> 1309773523007

9951281 -> 191375299 -> 3566721071 -> 73742690239 -> 2675036379181

10002547 -> 157769603 -> 4137197677 -> 64582279043 -> 1309807255657

10314503 -> 191012077 -> 4103955203 -> 64615521517 -> 1309774013183

10517383 -> 190809197 -> 4104158083 -> 64615318637 -> 1309774216063

10620521 -> 157151629 -> 4137815651 -> 64581661069 -> 1309807873631

10735111 -> 190591469 -> 4104375811 -> 64615100909 -> 1309774433791

10755149 -> 157017001 -> 4137950279 -> 64581526441 -> 1309808008259

11057171 -> 156714979 -> 4138252301 -> 64581224419 -> 1309808310281

11157709 -> 156614441 -> 4138352839 -> 64581123881 -> 1309808410819

11209831 -> 190116749 -> 4104850531 -> 64614626189 -> 1309774908511

11218439 -> 190108141 -> 4104859139 -> 64614617581 -> 1309774917119

11344973 -> 189981607 -> 4104985673 -> 64614491047 -> 1309775043653

11358311 -> 189968269 -> 4104999011 -> 64614477709 -> 1309775056991

11638741 -> 223242269 -> 3534854101 -> 73774557209 -> 2125248698327

11974663 -> 189351917 -> 4105615363 -> 64613861357 -> 1309775673343

12273673 -> 155498477 -> 4139468803 -> 64580007917 -> 1309809526783

12310027 -> 222570983 -> 3535525387 -> 73773885923 -> 2125249369613

12749039 -> 188577541 -> 4106389739 -> 64613086981 -> 1309776447719

12952397 -> 188374183 -> 4106593097 -> 64612883623 -> 1309776651077

12965189 -> 188361391 -> 4106605889 -> 64612870831 -> 1309776663869

13004203 -> 221876807 -> 3536219563 -> 73773191747 -> 2125250063789

13007791 -> 255427649 -> 2965797811 -> 74343613499 -> 2399557548979

13127501 -> 188199079 -> 4106768201 -> 64612708519 -> 1309776826181

13389781 -> 221491229 -> 3536605141 -> 73772806169 -> 2125250449367

13400141 -> 187926439 -> 4107040841 -> 64612435879 -> 1309777098821

14477381 -> 186849199 -> 4108118081 -> 64611358639 -> 1309778176061

14479681 -> 186846899 -> 3571249471 -> 82328096429 -> 2391573066049

14488711 -> 186837869 -> 4108129411 -> 64611347309 -> 1309778187391

14573831 -> 186752749 -> 4108214531 -> 64611262189 -> 1309778272511

14617507 -> 220263503 -> 3537832867 -> 73771578443 -> 2125251677093

14631361 -> 186695219 -> 3571401151 -> 82327944749 -> 2391573217729

14822671 -> 186503909 -> 4108463371 -> 64611013349 -> 1309778521351

15340261 -> 185986319 -> 4108980961 -> 64610495759 -> 1309779038941

15504007 -> 185822573 -> 4109144707 -> 64610332013 -> 1309779202687

15702091 -> 219178919 -> 3538917451 -> 73770493859 -> 2125252761677

15771181 -> 185555399 -> 4109411881 -> 64610064839 -> 1309779469861

16624771 -> 218256239 -> 3539840131 -> 73769571179 -> 2125253684357

19869973 -> 382783199 -> 9280893199 -> 265597013729 -> 5781716939017

20493037 -> 449268997 -> 7066923757 -> 147551898881 -> 4250494612207

20823839 -> 583155919 -> 12301745957 -> 365655376069 -> 7330926018349

21430561 -> 247004887 -> 5121704213 -> 97957510879 -> 3475455279367

21883871 -> 582095887 -> 12302805989 -> 365654316037 -> 7330927078381

24009061 -> 445752973 -> 7070439781 -> 147548382857 -> 4250498128231

24978637 -> 444783397 -> 7071409357 -> 147547413281 -> 4250499097807

25987837 -> 443774197 -> 9219902201 -> 265658004727 -> 6881167575791

27161677 -> 442600357 -> 7073592397 -> 147545230241 -> 4250501280847

31286917 -> 438475117 -> 7077717637 -> 147541105001 -> 4250505406087 -> 110098703882591

41155703 -> 1166803831 -> 28897967227 -> 795735753581 -> 18995473546369

49853357 -> 1023888451 -> 14008497071 -> 329588886589 -> 10665527391151

52407217 -> 1021334591 -> 16158534577 -> 327438849083 -> 9568165800883

57341617 -> 1016400191 -> 16163468977 -> 327433914683 -> 9568170735283

58642007 -> 1149317527 -> 24620486237 -> 800013234571 -> 16792172809829

61523771 -> 1012218037 -> 16167651131 -> 327429732529 -> 9568174917437

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Giorgos Kalogeropoulos wrote:

Q1.

5 primes: 178403 -> 2443027 -> 39500003 -> 900024079 -> 16279845089

6 primes: 698729 -> 9787021 -> 191539559 -> 4103427721 -> 64616048999 -> 1309773485701

7 primes: 36543351991 -> 1062968275769 -> 27524334046381 -> 605794363552577 -> 30919403028040867 -> 833771725427094341 -> 38365559431205702809

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Oscar wrote:
Here are my findings about puzzle 1082.

Q1) Base 2, first chain of n primes for n <= 8.
n = 5:
178403,
2443027,
39500003,
900024079,
16279845089.

n = 6:
698729,
9787021,
191539559,
4103427721,
64616048999,
1309773485701.

n = 7:
36543351991,
1062968275769,
27524334046381,
605794363552577,
30919403028040867,
833771725427094341,
38365559431205702809.

n = 8:
2749998982667,
67618745194981,
2043443580134909,
51999751948311019,
2542073633417094641,
117361762845694990837,
6080744245920714352097,
277261244726257998469633.

Q2) Base 2, two primes generating the same prime.
165624676339 -> 8080712531951
715380490249 -> 8080712531951

Q3) Base 10, first chain of n primes for n <= 5.
n = 3 (smaller than given examples):
127,
983,
1237.

n = 4:
71411,
84143,
138077,
2750809.

n = 5:
10000049932907,
478888838955977,
11076666716599567,
966701111061178201,
5477743333383266237.

***

Emmanuel wrote:

First of all,there might be some misunderstanding : In my opinion, we should have : 3 -> 0 (and further : a  ->  0  for all numbers  a  of the form  2^m - 1).
But there's no further trouble.

Q1 :
Five chains :
178403 -> 2443027 -> 39500003 -> 900024079 -> 16279845089
1001348477 -> 22620971629 -> 458415365509 -> 12735724167779 -> 303923624631691
Six Chains :
698729 -> 9787021 -> 191539559 -> 4103427721 -> 64616048999 -> 1309773485701
1107145729 -> 28957625329 -> 589517665277 -> 25798761401323 -> 888994912908283 -> 30636202478685161

Q2 :
I could not find two or more primes that generate the same prime.
The only pairs of primes I found that generate the same number all generated an even number.
That leads me to the conjecture that there are no exceptions, i. e. :
Two different primes cannot generate the same odd number.
But I have no Idea how to prove that.

Q3 :
Four chains :
71411 -> 84143 -> 138077 -> 2750809
11001111101 -> 77887777787 -> 811001111093 -> 2077887777793

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