looks like Shanks' idea is working, even when p is not a prime: Phi6(2^6^m) Phi10(2^10^m) Phi14(2^14^m) Phi15(2^15^m)
Phi6(2^6^m)
= Phi3(-2^6^m) = 4^6^m-2^6^m+1 = Phi(6^(m+1))(2) 0 3 1 37 * 109 2 33975937 * 138991501037953 3 10369 * 259201 * 1824045366145909775041 * 524833094849624730914401 * P(76) 4310500811590110373680321883148441662769860394543817514762656603162260887009 (0) 6^1 3 exception (1) 6^2 37-1 = 2^2 * 3^2 109-1 = 2^2 * 3^3 (2) 6^3 33975937-1 = 2^7 * 3^4 * 29 * 113 138991501037953-1 = 2^7 * 3^3 * 79 * 509081623 (3) 6^4 10369-1 = 2^7 * 3^4 259201-1 = 2^7 * 3^4 * 5^2 1824045366145909775041-1 = 2^6 * 3^6 * 5 * 19 * 23 * 17892733436939 524833094849624730914401-1 = 2^5 * 3^4 * 5^2 * 1061 * 7633625028357023 P(76)-1 = 2^5 * 3^4 * 101 * 117016168497363413 * 213610945241526331 * 658721721545608220012367588148904933
Phi10(2^10^m)
= Phi5(-2^10^m) = 16^10^m-8^10^m+4^10^m-2^10^m+1 = Phi(10^(m+1))(2) 0 11 1 5 * 101 * 8101 * 268501 2 4001 * 1074001 * 2020001 * 22624001 * 1481124532001 * P(85) 8877945148742945001146041439025147034098690503591013177336356694416517527310181938001 3 C(1205) (0) 10^1 11-1 = 2 * 5 (1) 10^2 5 exception 101-1 = 2^2 * 5^2 8101-1 = 2^2 * 3^4 * 5^2 268501-1 = 2^2 * 3 * 5^3 * 179 (2) 10^3 4001-1 = 2^5 * 5^3 1074001-1 = 2^4 * 3 * 5^3 * 179 2020001-1 = 2^5 * 5^4 * 101 22624001-1 = 2^8 * 5^3 * 7 * 101 1481124532001-1 = 2^5 * 5^3 * 2053 * 180361 P(85)-1 = 2^4 * 3^2 * 5^3 * 23 * 5701 * 3411721 * 18632456228623 * 59172164645028610815834244303800378546656478925516031549
Phi12(2^12^m)
= Phi6(4^12^m) = Phi3(-4^12^m) = 16^12^m-4^12^m+1 0 13 1 577 * 487824887233 2 1718990209 * 8148919324033 * C(152) 3 872256417178360321 * C(2063) (0) 12^1 13-1 = 2^2 * 3 (1) 12^2 577-1 = 2^6 * 3^2 487824887233-1 = 2^6 * 3^5 * 1091 * 28751 (2) 12^3 1718990209-1 = 2^7 * 3^3 * 13 * 38261 8148919324033-1 = 2^7 * 3^7 * 167 * 174311 (3) 12^4 872256417178360321-1 = 2^9 * 3^4 * 5 * 1231 * 1567 * 2180681
Phi14(2^14^m)
= Phi7(-2^14^m) = 64^14^m-32^14^m+16^14^m-8^14^m+4^14^m-2^14^m+1 = Phi(14^(m+1))(2) 0 43 1 197 * 19707683773 * 4981857697937 2 120737 * 18159793 * 400190449 * C(334) 3 614657 * 280010217471844569773539092961 * C(4921) (0) 14^1 43-1 = 2 * 3 * 7 (1) 14^2 197-1 = 2^2 * 7^2 19707683773-1 = 2^2 * 3 * 7^3 * 17 * 281651 4981857697937-1 = 2^4 * 7^2 * 127 * 337 * 148471 (2) 14^3 120737-1 = 2^5 * 7^3 * 11 18159793-1 = 2^4 * 3 * 7^3 * 1103 400190449-1 = 2^4 * 3 * 7^3 * 109 * 223 (3) 14^4 614657-1 = 2^8 * 7^4 280010217471844569773539092961-1 = 2^5 * 3 * 5 * 7^4 * 97 * 134936639 * 18562602237119
Phi15(2^15^m)
Phi15(x) = x^8-x^7+x^5-x^4+x^3-x+1 Phi15(2^15^m) = 256^15^m-128^15^m+32^15^m-16^15^m+8^15^m-2^15^m+1 0 151 1 115201 * 617401 * 1348206751 * 13861369826299351 2 8208001 * 8971209001 * P(525) 970295059340550810438151650519081760016369915666679773075505533374073727309076136591693767960086757310421013734311551302462059783237024870934353724032560786730638388324169656182168970960945024110555649325990018893771759838483563627533169740269044556094267066180504212230697412080098904985404293546183520918413502860296641731928133549616148232852834073964729649449901727627198274082695497973382950481692211920204819964986610470518699195044470454663226109071334066309162789371616658778666668501540911528778054844182967540943001 3 583552569813751 * C(8114) (0) 2*15^1 151-1 = 2 * 3 * 5^2 (1) 2*15^2 115201-1 = 2^9 * 3^2 * 5^2 617401-1 = 2^3 * 3^2 * 5^2 * 7^3 1348206751-1 = 2 * 3^2 * 5^3 * 11 * 19 * 47 * 61 13861369826299351-1 = 2 * 3^2 * 5^2 * 41 * 1933 * 388667231 (2) 2*15^3 8208001-1 = 2^7 * 3^3 * 5^3 * 19 8971209001-1 = 2^3 * 3^3 * 5^3 * 13 * 61 * 419 P(525)-1 = 2^3 * 3^3 * 5^3 * 90885856991 * C(510) (3) 2*15^4 583552569813751-1 = 2 * 3^4 * 5^4 * 17 * 113 * 3000251
Phi18(2^18^m)
= Phi6(8^18^m) = Phi3(-8^18^m) = 64^18^m-8^18^m+1 0 3 * 19 1 3618757 * 106979941 * 168410989 * 4977454861 2 139969 * C(581) 3 C(10534) (0) 18^1 3 exception 19-1 = 2 * 3^2 (1) 18^2 3618757-1 = 2^2 * 3^6 * 17 * 73 106979941-1 = 2^2 * 3^4 * 5 * 66037 168410989-1 = 2^2 * 3^4 * 519787 4977454861-1 = 2^2 * 3^4 * 5 * 7 * 31 * 14159 (2) 18^3 139969-1 = 2^6 * 3^7
Phi21(2^21^m)
Phi21(x) = x12-x11+x9-x8+x6-x4+x3-x+1 Phi21(2^21^m) = 4096^21^m-2048^21^m+512^21^m-256^21^m+64^21^m-16^21^m+8^21^m-2^21^m+1 0 7 * 337 1 126127 * 309583 * 5828257 * 4487533753346305838985313 * 7086423574853972147970086088434689 2 C(1594) 3 2793967688877020839 * C(33436) (0) 2*21^1 7 exception 337-1 = 2^4 * 3 * 7 (1) 2*21^2 126127-1 = 2 * 3^2 * 7^2 * 11 * 13 309583-1 = 2 * 3^5 * 7^2 * 13 5828257-1 = 2^5 * 3^2 * 7^3 * 59 4487533753346305838985313-1 = 2^5 * 3^3 * 7^2 * 3049 * 80239 * 433266363247 7086423574853972147970086088434689-1 = 2^10 * 3^2 * 7^2 * 37 * 424118129700210343552861261 (3) 2*21^4 2793967688877020839-1 = 2 * 3^4 * 7^4 * 13 * 170633 * 3238231
Phi30(2^30^m)
Phi30(x) = x^8+x^7-x^5-x^4-x^3+x+1 Phi30(2^30^m) = 256^30^m+128^30^m-32^30^m-16^30^m-8^30^m+2^30^m+1 0 331 1 695701 * 307116398490301 * 413150254353901 * 6269989892198401 * 3192261504216112476901 2 2140830001 * 5522688001 * 10896550812001 * 1566364275534049506001 * C(2115) 3 (0) 30^1 331-1 = 2 * 3 * 5 * 11 (1) 30^2 695701-1 = 2^2 * 3^2 * 5^2 * 773 307116398490301-1 = 2^2 * 3^2 * 5^2 * 7 * 1063 * 1721 * 26647 413150254353901-1 = 2^2 * 3^2 * 5^2 * 7 * 47 * 719 * 1940621 6269989892198401-1 = 2^12 * 3^3 * 5^2 * 17 * 133399499 3192261504216112476901-1 = 2^2 * 3^2 * 5^2 * 37 * 43 * 97 * 179 * 171929 * 746813 (2) 30^3 2140830001-1 = 2^4 * 3^5 * 5^4 * 881 5522688001-1 = 2^11 * 3^3 * 5^3 * 17 * 47 10896550812001-1 = 2^5 * 3^3 * 5^3 * 7^2 * 41 * 50221 1566364275534049506001-1 = 2^4 * 3^3 * 5^3 * 13397599 * 2165070461
Phi2(2^6^m) = 2^6^m+1 0 3(2) 1 5(4) * 13(12) 2 17(8) * 241(24) * 433(72) * 38737(72) 3 97(48) * 257(16) * 577(144) * 673(48) * 209924353(432) * 4261383649(432) * 487824887233(144) * 24929060818265360451708193(432) 0 Phi2(2) 1 Phi4(2) Phi12(2) 2 Phi8(2) Phi24(2) Phi72(2) 3 Phi16(2) Phi48(2) Phi144(2) Phi432(2) 4 Phi32(2) Phi96(2) Phi288(2) Phi864(2) Phi2592(2) 0 Phi2(2^2^0) 1 Phi2(2^2^1) Phi3(-2^2^1) 2 Phi2(2^2^2) Phi3(-2^2^2) Phi3(-8^2^2) 3 Phi2(2^2^3) Phi3(-2^2^3) Phi3(-8^2^3) Phi3(-512^2^3) 4 Phi2(2^2^4) Phi3(-2^2^4) Phi3(-8^2^4) Phi3(-512^2^4) Phi6(-134217728^2^4)
Phi3(2^6^m) = 4^6^m+2^6^m+1 0 7 1 3 * 19 * 73 2 3 * 87211 * 246241 * 262657 * 279073 3 3 * 163 * 1297 * 2593 * 3889 * 71119 * 135433 * 3618757 * 30433969 * 97685839 * 106979941 * 168410989 * 272010961 * 1164777409 * 4977454861 * 3718266498433 * 134921168163073 * 1174029487714513 4 3 * 487 * 1459 * 2917 * 4861 * 139483 * 10194337 * 26232337 * 8179496641 * 13655624113 * 3333950193493 * 16753783618801 * 26129603777437 * 192971705688577 * 500455248092353 * 3712990163251158343 * 10429407431911334611 * 918125051602568899753 * 15778453094691989880197773477 * C(242) * P(359) 1 Phi9(2) Phi18(2) 2 27 54 108 3 81 162 324 648 81 7 * 73 * 262657 : 2593 * 71119 * 97685839
Phi15(-2^15^m)
Phi15(x) = x^8-x^7+x^5-x^4+x^3-x+1 Phi30(x) = x^8+x^7-x^5-x^4-x^3+x+1 = Phi15(-x) Phi30(2^15^m) = 256^15^m+128^15^m-32^15^m-16^15^m-8^15^m+2^15^m+1 0 331 1 4714696801 * 281941472953710177758647201 2 5474391804001 * 740140319718001 * C(515) 3 405001 * C(8123) (0) 2*15^1 331-1 = 2 * 3 * 5 * 11 (1) 2*15^2 4714696801-1 = 2^5 * 3^3 * 5^2 * 11 * 19843 281941472953710177758647201-1 = 2^5 * 3^2 * 5^2 * 11 * 131 * 947 * 28695414871783663 (2) 2*15^3 5474391804001-1 = 2^5 * 3^4 * 5^3 * 7 * 31 * 77863 740140319718001-1 = 2^4 * 3^3 * 5^3 * 3187 * 4300691 (3) 2*15^4 405001-1 = 2^3 * 3^4 * 5^4
2 : 3 4 3 : 5 8 3 * 5 : 17 16 3 * 5 * 17 : 257 32 3 * 5 * 17 * 257 : 65537 64 3 * 5 * 17 * 257 * 65537 : 641 * 6700417 128 3 * 5 * 17 * 257 * 641 * 65537 * 6700417 : 274177 * 67280421310721 256 3 * 5 * 17 * 257 * 641 * 65537 * 274177 * 6700417 * 67280421310721 : 59649589127497217 * 5704689200685129054721 3 : 7 9 7 : 73 27 7 * 73 : 262657 81 7 * 73 * 262657 : 2593 * 71119 * 97685839 243 7 * 73 * 2593 * 71119 * 262657 * 97685839 : 487 * 16753783618801 * 192971705688577 * 3712990163251158343 5 : 31 25 31 : 601 * 1801 125 31 * 601 * 1801 : 269089806001 * 4710883168879506001 6 3 * 7 : (3) 36 3^3 * 5 * 7 * 13 * 19 * 73 : 37 * 109 216 3^4 * 5 * 7 * 13 * 17 * 19 * 37 * 73 * 109 * 241 * 433 * 38737 * 87211 * 246241 * 262657 * 279073 : 33975937 * 138991501037953 10 3 * 31 : 11 100 3 * 5^2 * 11 * 31 * 41 * 251 * 601 * 1801 * 4051 : (5) * 101 * 8101 * 268501