Shanks(7+) Phi7(2^7^m) = 64^7^m+32^7^m+16^7^m+8^7^m+4^7^m+2^7^m+1 64^7^3+32^7^3+16^7^3+8^7^3+4^7^3+2^7^3+1 64^7^4+32^7^4+16^7^4+8^7^4+4^7^4+2^7^4+1 64^7^5+32^7^5+16^7^5+8^7^5+4^7^5+2^7^5+1 factors: k*(2*7^(m+1))+1 see 7-Proths 0 127 1 4432676798593 2 6073159 * 1428389887 * 62228099977 * 58961804474844164724814095915114338093146118248375213688557057 3 14407 * 1493423 * 4725169 * C(603) #1315 4 C(4337) #423 5 8470729 (#1) * 900744038503 (#1) * 226948319644313 (#30) * C(30324) #133
never 3^2 except prime never 5 (others?) (0) 2*7^1 127-1 = 2 * 3^2 * 7 (1) 2*7^2 4432676798593-1 = 2^7 * 3^2 * 7^2 * 43 * 337 * 5419 (2) 2*7^3 6073159-1 = 2 * 3 * 7^3 * 13 * 227 1428389887-1 = 2 * 3 * 7^3 * 11 * 63097 62228099977-1 = 2^3 * 3 * 7^4 * 41 * 26339 [<<<<< extra 7] 58961804474844164724814095915114338093146118248375213688557057-1 = 2^9 * 3 * 7^3 * 191 * 585938538228697659132042653540557810857773837273303717 (3) 2*7^4 14407-1 = 2 * 3 * 7^4 1493423-1 = 2 * 7^4 * 311 [<<<<<< no 3] 4725169-1 = 2^4 * 3 * 7^4 * 41 (5) 2*7^6 8470729-1 = 2^3 * 3^2 * 7^6 900744038503-1 = 2 × 3 × 7^6 × 11 × 311 × 373 226948319644313-1 = 2^3 × 7^6 × 173 × 1393807
7-Proth numbers k*7^n+1 where k is even but not divisible by 7 dual: k+7^n 2 4 6 8 10 12 16 18 20 22 24 26 30 32 34 36 38 40 44 46 48 50 52 54 58 60 62 64 66 68 72 74 76 78 80 82 86 88 90 92 94 96 100 102 104 106 108 110
Phi(7^(m+1),x) = Phi7(x^7^m) rank prime digits who when comment 111357 30*7^1944+1 1645 K Dec 1994 Divides Phi(7^1944,2)=Phi7(2^7^1943) 74073 10*7^29447+1 24887 g151 Aug 2000 Divides Phi(7^29446,2)=Phi7(2^7^29445) 68890 72*7^40122+1 33909 g151 May 2002 Divides Phi(7^40121,2)=Phi7(2^7^40120) 60485 10*7^70115+1 59256 p67 Aug 2004 Divides Phi(7^70114,2)=Phi7(2^7^70113) 807 4*7^1640811+1 1386647 A2 Sep 2024 31193 4*7^266782+1 225458 p264 Dec 2009 Generalized Fermat (2*7^133391)^2^1+1 32257 4*7^254621+1 215181 p264 Dec 2009 56410 4*7^94133+1 79553 p67 Mar 2004 80966 4*7^16474+1 13923 g59 Nov 1998 Generalized Fermat (2*7^8237)^2^1+1 28853 10*7^297576+1 251482 p264 Dec 2009 56119 10*7^97249+1 82186 p67 Jul 2006 56301 10*7^95122+1 80389 p67 May 2006 56587 10*7^92289+1 77995 g375 Apr 2006 60485 10*7^70115+1 59256 p67 Aug 2004 Divides Phi(7^70114,2) 74073 10*7^29447+1 24887 g151 Aug 2000 Divides Phi(7^29446,2) 80689 10*7^17005+1 14372 g151 Jul 2000 85989 10*7^11393+1 9630 gk Apr 1999 73358 12*7^31642+1 26742 g151 Oct 2001 88538 12*7^8794+1 7433 g103 Jan 1999 87742 16*7^9332+1 7888 gk Apr 1999 Generalized Fermat (2*7^2333)^2^2+1 90941 18*7^7732+1 6536 g103 Dec 1998 8173 30*7^670289+1 566462 L3610 Dec 2014 56605 76*7^92068+1 77809 p67 Mar 2006 56909 60*7^89320+1 75486 p67 Jul 2005 57000 30*7^88677+1 74943 g375 Jun 2005 57236 48*7^86382+1 73003 g375 Apr 2005 57717 90*7^83259+1 70364 g375 Mar 2005 57830 64*7^82550+1 69765 g375 Feb 2005 Generalized Fermat (2^3*7^41275)^2^1 58053 66*7^81328+1 68732 g375 Jan 2005 58192 30*7^80139+1 67727 g151 May 2002 58886 66*7^76229+1 64423 p67 Nov 2004 59114 72*7^74654+1 63092 p67 Oct 2004 59170 64*7^74150+1 62666 p67 Oct 2004 Generalized Fermat (2^3*7^37075)^2^1 59363 94*7^72761+1 61493 p67 Sep 2004 59378 78*7^72634+1 61385 p67 Aug 2004 59415 72*7^72402+1 61189 p67 Aug 2004 60468 48*7^70276+1 59392 p67 Aug 2004 60505 34*7^70049+1 59200 p67 Aug 2004 60589 40*7^69939+1 59107 p67 Jul 2004 60953 90*7^68084+1 57540 p67 Jul 2004 62866 96*7^62009+1 52406 p67 Jun 2004 63153 60*7^61338+1 51839 p67 May 2004 63362 72*7^60910+1 51477 p67 May 2004 68765 76*7^40867+1 34539 g151 May 2002 73090 24*7^32239+1 27247 g151 Oct 2001 75333 34*7^26203+1 22146 g151 Jul 2000 75887 72*7^23919+1 20216 g103 Jan 1999 86783 24*7^10363+1 8760 gk Apr 1999 86786 30*7^10359+1 8756 gk Apr 1999 86798 24*7^10335+1 8736 gk Apr 1999 88978 66*7^8532+1 7213 g103 Dec 1998 40925 964*7^200026+1 169045 g107 Jan 2004 87319 202*7^9900+1 8369 gk Nov 1998 Generalized Cullen 9898*7^9898+1 108233 282*7^2366+1 2025 CR Aug 1996 4821 1084*7^862557+1 728949 L5211 Oct 2021 87235 7120*7^10000+1 8455 C May 1994 111243 1980*7^1980+1 1677 D Dec 1989 Generalized Cullen 5886 353938*7^777777+1 657304 L4789 Oct 2020 87247 823542*7^9982+1 8442 g138 May 1999 62278 19771230*7^63684+1 53827 g358 Oct 2004 58530 2603102760*7^77777+1 65739 A25 Mar 2024 Twin (p+2) 65197 22835841624*7^54321+1 45917 p296 Nov 2010 Twin (p+2) 82449 2^1001*3^1003*7^14289+1 12856 p21 8 Feb 2000 Arlin E. Anderson starship1@gmail.com