Shanks(5+)
Phi5(2^5^m) = 16^5^m+8^5^m+4^5^m+2^5^m+1

factors: k*(2*5^(m+1))+1  see 5-Proths

16^5^4+8^5^4+4^5^4+2^5^4+1
16^5^5+8^5^5+4^5^5+2^5^5+1
16^5^6+8^5^6+4^5^6+2^5^6+1

0 31
1 601 * 1801
2 269089806001 * 4710883168879506001
3 1277297679372570001 * 573759820507018639639785001 * 18152902839291497575027462639977160832701118299213751 * 246053469753590746981511859818675718355368494592178751
4 2839015665925001 (#2) * 8959931591578725001 (#103) * C(719) #1141
5 C(3763) #417
6 C(18815) #255
7 [need a primality test]
8
9



(0) 2*5^1
31-1 = 2 * 3 * 5

(1) 2*5^2
601-1 = 2^3 × 3 × 5^2
1801-1 = 2^3 × 3^2 × 5^2

(2) 2*5^3
269089806001-1 = 2^4 × 3 × 5^3 × 41 × 107 × 10223
4710883168879506001-1 = 2^4 × 3^2 × 5^3 × 41^2 × 184711 × 842887

(3) 2*5^4
1277297679372570001-1 = 2^4 × 3^2 × 5^4 × 14192196437473
573759820507018639639785001-1 = 2^3 × 3 × 5^4 × 6143 × 9283 × 19379891 × 34611361
18152902839291497575027462639977160832701118299213751-1 = 2 × 5^4 × 569 × 1319 × 9049 × 24077 × 6198377 × 22580392210333 × 634551066796877
246053469753590746981511859818675718355368494592178751-1 = 2 × 5^4 × 7 × 11 × 23201 × 3229181 × 16064768673841 × 2124003297492931256519479

(4) 2*5^5
2839015665925001-1 = 2^3 × 5^5 × 17 × 61 × 181 × 605021
8959931591578725001-1 = 2^3 × 3 × 5^5 × 473633 × 252232751



divisors of Phi5(2^5^m)
m digits             k n
0  2                 6 1
1  3                24 2
1  4                72 2



Phi(5^(m+1),x) = Phi5(x^5^m)

86672	38*5^12727 + 1	8898	gk	Jan 2000	divides Phi(5^12727,2)=Phi5(2^5^12721)
58695	22*5^93078 + 1	65061	p67	Nov 2004	divides Phi(5^93077,2)=Phi5(2^5^93076)


33803	96436*5^289308 + 1	202223	L2777	Dec 2011	Generalized Cullen (**)
2588	37292*5^1487989 + 1	1040065	L3553	Dec 2013	(**)
2386	59912*5^1500861 + 1	1049062	L3772	Jan 2014	(**)
931	24032*5^1768249 + 1	1235958	L3925	Jul 2014	(**)
710	77072*5^2139921 + 1	1495746	L4340	Mar 2016	(**)
706	92158*5^2145024 + 1	1499313	L4348	Mar 2016	(**)
578	81556*5^2539960 + 1	1775361	L4809	Jun 2018	(**)
30881	109104*5^327312 + 1	228787	L2841	Feb 2012	Generalized Cullen (**)
919	133778*5^1785689 + 1	1248149	L3903	Aug 2014	(**)
900	109208*5^1816285 + 1	1269534	L3523	Oct 2014	(**)
785	144052*5^2018290 + 1	1410730	L4146	May 2015	(**)
731	154222*5^2091432 + 1	1461854	L3523	Nov 2015	(**)
482	138514*5^2771922 + 1	1937496	L4937	Apr 2019	(**)
282	118568*5^3112069 + 1	2175248	L690	May 2020	(**)
91	2805222*5^5610444 + 1	3921539	L4972	Sep 2019	Generalized Cullen (**)






5-Proth numbers
k*5^n+1 where k is even but not divisible by 5

2
4
6
8

12
14
16
18
22
24
26
28
32
34
36
38
42
44
46
48
52
54
56
58
62
64
66
68
72
74
76
78
82
84
86
88
92
94
96
98

102
104
106
108