Cycles in generalized Collatz-transformation.

We consider the generalized Collatz-transformation for some given prime p

                        a_k+1= {p*a_k + 1}_p-1     

                                               where {n}_Q means: remove all primefactors q_k <Q    from some given n

                        (see for instance https://math.stackexchange.com/questions/4653947/collatz-conjecture-generalization )

Here we search for existence of cycles; the range of basic primes is p=2..97, and p=2⁵-1=31,2⁷-1=127,2¹³-1,2¹⁷-1,2¹⁹-1

Note_1: a₁ with absolute values from 2.. 10⁶ are used.

Note_2: Cycles with larger first element might occur when some valid a₁ leads to higher values than 10⁶ and after some iterations a cycle begins in that larger numbers

prime p=5

cycles:

  - - - - - - -

-7(2): -7 -17 -7

 

prime p=7

cycles:

  - - - - - - -

 -11(2): -11 -19 -11

-509(4): -509 -1781 -6233 -4363 -509

-701(4): -701 -2453 -1717 -2003 -701

-961(4): -961 -1121 -3923 -1373 -961

 

prime p=11

cycles:

 17(3): 17 47 37 17

  - - - - - - -

-13(2): -13 -71 -13

-17(2): -17 -31 -17

 

prime p=13

cycles:

    19( 4): 19 31 101 73 19

  - - - - - - -

   -19( 2): -19 -41 -19

 -9959(19): -9959 -64733 -105191 -683741 -1111079 -7222013 -11735771 -76282511 -495836321 -1611468043 -1163838031 -120078527 -31220417 -20293271 -131906261 -107173837 -11610499 -25156081 -3893203 -9959

-99503(17): -99503 -646769 -2101999 -4554331 -365471 -2375561 -1102939 -796567 -345179 -2243663 -14583809 -47397379 -102694321 -111252181 -30130799 -2543509 -688867 -99503

 

prime p=17

cycles:

 43(22): 43 61 173 1471 521 4429 4183 2963 257 437 743 1579 2237 3803 2309 19627 5561 47269 14881 3833 32581 263 43

  - - - - - - -

-29( 2): -29 -41 -29

 

prime p=19

cycles:

46063(16): 46063 437599 4157191 7898663 25012433 39603019 376228681 32492477 25723211 16291367 1563313 675067 6413137 30462401 1378061 72731 46063

  - - - - - - -

  -23( 2): -23 -109 -23

  -83( 6): -83 -197 -1871 -8887 -14071 -22279 -83

 

prime p=23

cycles:

 179(4): 179 2059 877 1681 179

  - - - - - - -

 -31(2): -31 -89 -31

 

prime p=29

cycles:

  - - - - - - -

  -31( 2): -31 -449 -31

  -43( 2): -43 -89 -43

-4337(11): -4337 -10481 -8443 -17489 -8453 -5107 -74051 -357913 -2594869 -5879 -5683 -4337

 

prime p=31

cycles:

 67(30): 67 1039 3221 8321 2687 13883 10247 4813 3391 52561 101837 29231 151027 334417 197 509 263 151 2341 18143 93739 41513 1849 1433 617 797 71 367 5689 4409 67

  - - - - - - -

-37( 2): -37 -191 -37

-41( 2): -41 -127 -41

 

prime p=37

cycles:

 2173(17): 2173 5743 7589 46799 16033 42373 783901 1318379 2032501 37601269 975629 6016379 545603 420569 864503 18133 2819 2173

  - - - - - - -

  -41( 2): -41 -379 -41

 

prime p=41

cycles:

  - - - - - - -

    -43(2): -43 -881 -43

    -53(2): -53 -181 -53

    -71(2): -71 -97 -71

 -33493(6): -33493 -343303 -413983 -8486651 -3866141 -125803 -33493

 -44917(6): -44917 -460399 -555187 -11381333 -3535111 -59159 -44917

 -76081(6): -76081 -77983 -1598651 -128519 -125459 -857303 -76081

 -82933(4): -82933 -850063 -17426291 -821239 -82933

 -87931(4): -87931 -360517 -3695299 -10821947 -87931

-124447(4): -124447 -2551163 -2490421 -176047 -124447

 

prime p=43

cycles:

174569(13): 174569 208513 448303 1927703 2763041 3300299 23652143 6780281 8098669 21765173 236339 1693763 2427727 174569

  - - - - - - -

    -71(2): -71 -109 -71

 -94769(4): -94769 -2037533 -6258137 -727297 -94769

-138113(4): -138113 -2969429 -246497 -1059937 -138113

 

prime p=47

cycles:

   2243(14): 2243 52711 412903 294037 20939 492067 154181 1811627 8514647 40018841 171739 122299 73693 41233 2243

  - - - - - - -

    -71(2): -71 -139 -71

 -63979(8): -63979 -751753 -3533239 -2965397 -7742981 -5513941 -379993 -1785967 -63979

-128437(4): -128437 -3018269 -1390771 -1257043 -128437

-130597(4): -130597 -3069029 -1045249 -1444903 -130597

-178813(4): -178813 -840421 -858691 -593507 -178813

-197629(4): -197629 -273193 -1284007 -7543541 -197629

-239017(4): -239017 -244213 -1147801 -2074871 -239017

-256691(4): -256691 -1005373 -363481 -502459 -256691

-258337(4): -258337 -263953 -1240579 -857459 -258337

-281873(4): -281873 -441601 -451201 -623719 -281873

 

 

prime p=53

cycles:

   19559(14): 19559 259157 6867661 60664339 22327847 295843973 223996151 102343069 816649 7213733 332459 200231 2653061 10043731 19559

  - - - - - - -

     -59( 2): -59 -521 -59

     -89( 2): -89 -131 -89

-1163411(11): -1163411 -3425599 -12968339 -114553661 -4864859 -4774769 -7029521 -1478431 -2304613 -15268061 -4214621 -1163411

 

prime p=59

cycles:

 73(7): 73 359 89 101 149 157 193 73

  - - - - - - -

-71(2): -71 -349 -71

-79(2): -79 -233 -79

 

prime p=61

cycles:

   97(3): 97 269 547 97

  199(7): 199 607 9257 10457 2593 79087 1206077 199

26833(6): 26833 818407 290249 590173 947383 14447591 26833

  - - - - - - -

  -71(2): -71 -433 -71

 

prime p=67

cycles:

 181(3): 181 379 12697 181

  - - - - - - -

 -89(2): -89 -271 -89

-101(2): -101 -199 -101

-109(6): -109 -1217 -691 -643 -359 -859 -109

 

prime p=71

cycles:

 306511(7): 306511 3627047 6776851 80192737 79078949 280730269 22146499 306511

  - - - - - - -

    -73(2): -73 -2591 -73

    -79(2): -79 -701 -79

    -83(2): -83 -491 -83

   -101(2): -101 -239 -101

   -107(2): -107 -211 -107

   -113(2): -113 -191 -113

-173021(6): -173021 -409483 -7268323 -129012733 -4579952021 -73736189 -173021

 

prime p=73

cycles:

 14929( 8): 14929 544909 19889179 51853931 147473 119617 17257 89983 14929

140729(12): 140729 1712203 6249541 228108247 25383997 132359413 193244743 44083957 123774187 2258878913 6913969 6820537 140729

  - - - - - - -

   -353(4): -353 -3221 -2027 -14797 -353

  -1429(9): -1429 -8693 -158647 -386041 -2621 -1543 -18773 -342607 -4933 -1429

-433003(4): -433003 -5268203 -14791493 -269944747 -433003

-446111(4): -446111 -16283051 -45717797 -4069999 -446111

-544421(4): -544421 -9935683 -3267139 -3057707 -544421

-557671(4): -557671 -6784997 -24765239 -1695931 -557671

-610411(4): -610411 -7426667 -1322309 -24132139 -610411

-713239(4): -713239 -8677741 -1287551 -3615047 -713239

-727357(4): -727357 -884951 -32300711 -28755511 -727357

-744397(4): -744397 -905683 -11019143 -30938363 -744397

-816869(4): -816869 -14907859 -13952227 -828059 -816869

-877739(4): -877739 -2464421 -1215559 -4929767 -877739

-929371(4): -929371 -11307347 -2230909 -1043951 -929371

 

prime p=79

cycles:

  - - - - - - -

   -131( 2): -131 -199 -131

   -137(11): -137 -773 -1607 -2267 -44773 -45347 -47137 -206879 -4751 -317 -659 -137

  -1307( 4): -1307 -25813 -2111 -1489 -1307

  -8741( 4): -8741 -345269 -21821 -861929 -8741

-191627(13): -191627 -3784633 -4530091 -9941033 -56095829 -443157049 -1166980229 -542302577 -578944643 -85971103 -141494107 -7961563 -52413623 -191627

 

prime p=83

cycles:

     89(21): 89 1847 76651 151477 261929 418079 17350279 240012193 135131 5607937 38788231 16259713 110801 19001 394271 1487477 7716287 320225911 340753213 522589 35437 89

   4049(22): 4049 84017 1743353 76157 790129 5465059 75599983 9093911 53913901 62150747 368465143 339806743 49480631 6867713 28501009 17921089 5389313 604477 2090483 70247 47791 661109 4049

  - - - - - - -

    -89( 2): -89 -1231 -89

-747613( 4): -747613 -4432277 -12262633 -9601873 -747613

 

prime p=89

cycles:

  - - - - - - -

 -113(2): -113 -419 -113

-1489(7): -1489 -3313 -36857 -1847 -27397 -3607 -12347 -1489

 

prime p=97

cycles:

 1109(22): 1109 17929 66889 170743 108961 480419 57961 401587 36749 16057 11981 193693 9394111 401069 6483949 16551133 802729951 12942953 4866149 864499 20964101 6311 1109

  - - - - - - -

 -109( 2): -109 -881 -109

 

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 p  from Mersenne-primes:

 

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prime p=2^7-1 (=127)

cycles:

  - - - - - - -

  -131(2):  -131 -4159 -131

  -139(2):  -139 -1471 -139

  -191(2):  -191  -379 -191

  -239(2):  -239  -271 -239

 

prime p=2^13-1 

cycles:

 118583(17):  118583 4905623 6696992999 2148651377 122219468257 1646545500811 4156807 17024203069 2045251501 12729981037 26067818668517 114150635177 3911904863 4513513 1782211 155298833 4379623 118583...

  - - - - - - -

    -8231(2):     -8231  -1685503   -8231

    -8233(2):     -8233  -1605631   -8233

    -8263(2):     -8263   -940031   -8263

    -8287(2):     -8287   -707071   -8287

    -8447(2):     -8447   -270271   -8447

    -8737(2):     -8737   -131071   -8737

    -8831(2):     -8831   -113023   -8831

    -8971(2):     -8971    -94207   -8971

    -9151(2):     -9151    -78079   -9151

    -9343(2):     -9343    -66431   -9343

   -11519(2):    -11519    -28351  -11519

   -12671(2):    -12671    -23167  -12671

   -12799(2):    -12799    -22751  -12799

   -15359(2):    -15359    -17551  -15359

 

prime p=2^17-1

cycles:

  132409(7): 132409, 4472933, 59507897, 2074903, 5912165459, 1305137, 1069267, 132409,

  - - - - - - -

 -136511(2): -136511 -3289087 -136511

 -141311(2): -141311 -1808767 -141311

 -174761(2): -174761  -524287 -174761

 -235927(2): -235927  -294911 -235927

 

prime p=524287 \\ =(2^19-1)

cycles:

  - - - - - - -

 -524351(2):  -524351 -4295475199  -524351

 -525199(2):  -525199  -301924351  -525199

 -525439(2):  -525439  -239132671  -525439

 -538651(2):  -538651   -19660799  -538651

 -546479(2):  -546479   -12910591  -546479

 -561079(2):  -561079    -7995391  -561079

 -562591(2):  -562591    -7700479  -562591

 -653311(2):  -653311    -2654719  -653311

 -657439(2):  -657439    -2588671  -657439

-1047551(2): -1047551    -1049599 -1047551

 

(valid 1<a₁ <1 000 000 : for p=2^19-1 number of admissible initial values: 35110)

 

 

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Gottfried Helms, 10'2024