Minimal set of the strings for primes with at least two digits

[sweety439]

The smallest prime of the form {1}2 in base b (let the length of this prime be n) is minimal prime (start with b+1) in base b if and only if the length of the smallest prime of the form {1} in base b is > n-1 (in the case that there is no prime of the form {1} in base b, i.e. b is in https://oeis.org/A096059, we let "the length of the smallest prime of the form {1} in base b" be infinity (like http://gladhoboexpress.blogspot.com/...-derbread.html and http://chesswanks.com/seq/a269254.txt, also see the thread https://mersenneforum.org/showthread.php?t=27636, e.g. "the smallest n>=1 such that k*2^n+1 is prime" should be infinity (instead of 0) for k = 78557 and 271129)), and infinity is > any finite number

[sweety439]

(n)111...111 (base 9) is always composite if (not proven "only if") ....

* n == 5, 6 mod 10 (then (n)111...111 (base 9) has covering set {2,5})
* n is triangular number (then (n)111...111 (base 9) has difference-of-squares factorization)

(n)111...111 (base 25) is always composite if (not proven "only if") ....

* n == 13, 14 mod 26 (then (n)111...111 (base 25) has covering set {2,13})
* n is generalized pentagonal number (then (n)111...111 (base 25) has difference-of-squares factorization)

[sweety439]

Newest condensed table of the problem: (no data available for bases 29, 31, 33, 35)

b	number of quasi-minimal primes base b	base-b form of largest known quasi-minimal prime base b	length of largest known quasi-minimal prime base b	length of largest known quasi-minimal prime base b when written in decimal	algebraic ((a×bn+c)/d) form of largest known quasi-minimal prime base b	number of unsolved families in base b	searching limit of length for the unsolved families in base b (if there are different searching limits for the unsolved families in base b, choose the lowest searching limit)	equivalent searching limit of length for the unsolved families in base b in decimal (if there are different searching limits for the unsolved families in base b, choose the lowest searching limit)
2	1	11	2	1	3	0	–	–
3	3	111	3	2	13	0	–	–
4	5	221	3	2	41	0	–	–
5	22	109313	96	67	595+8	0	–	–
6	11	40041	5	4	5209	0	–	–
7	71	3161	17	15	(717−5)/2	0	–	–
8	75	42207	221	200	(4×8221+17)/7	0	–	–
9	151	30115811	1161	1108	3×91160+10	0	–	–
10	77	502827	31	31	5×1030+27	0	–	–
11*	1068	5762668	62669	65263	(57×1162668−7)/10	0	–	–
12	106	403977	42	45	4×1241+91	0	–	–
13*	3196~3197	95197420	197421	219916	(113×13197420−5)/12	1	300000	334183.0056920510
14	650	4D19698	19699	22578	5×1419698−1	0	–	–
15	1284	715597	157	185	(15157+59)/2	0	–	–
16*	2347	3116137AF	116139	139845	(16116139+619)/5	0	–	–
17*	10412~10428	F701867671	186770	229811	262×17186768+1	17	100000	123044.8921378274
18	549	C06268C5	6271	7872	12×186270+221	0	–	–
19*	31412~31435	1E701228961	122900	157158	634×19122897+1	23	100000	127875.3600952829
20	3314	G06269D	6271	8159	16×206270+13	0	–	–
21*	13383~13394	CF4791470K	479150	633542	(51×21479149−1243)/4	11	50000	26444.3858946784
22*	8003	BK220015	22003	29538	(251×2222002−335)/21	0	–	–
23*	65149~65272	9E800873	800874	1090573	(106×23800873−7)/11	125	20000	66110.9647366960
24	3409	N00N8129LN	8134	11227	13249×248131−49	0	–	–
26*	25255~25259	M0611862BB	61190	86583	22×2661189+1649	4	100000	141497.3347970818
28*	25528~25529	O4O945359	94538	136812	(6092×2894536−143)/9	1	543203	786100.5840991875
30*	2619	OT34205	34206	50527	25×3034205−1	0	–	–
32*	168833~169017	NU06618631	18871	28404	766×32661864+1	184	20000	30102.9995663981
34*	184750~184834	U19778KCF	19781	30295	(10×3419781−134067)/11	84	20000	30629.5783408451
36*	35260~35263	P81993SZ	81995	127609	(5×3681995+821)/7	3	100000	155630.2500767287

A curious:

9453 (a song of a Taiwanese band 911, recently a member of 911 sent an Instagram message to Lionel Messi and Lionel Messi replied him and said that his favorite song of 911 is 9453) - 8003 (number of minimal primes (start with b+1) in base b = 22) = 1450 (台灣網路用語，用於諷刺偏袒民進黨的網絡水軍, just like 426 (死阿六, 阿陸仔))

[sweety439]

Proth primes base b: Primes of the form k*b^n+1
Riesel primes base b: Primes of the form k*b^n-1
Dual Proth primes base b: Primes of the form b^n+k
Dual Riesel primes base b: Primes of the form b^n-k

OEIS sequences:

A = Smallest n>=1 making the number prime, for b = 2
B = Smallest prime in the form with n>=1, for b = 2
C = Smallest n>=1 making the number prime, for b = 2 and odd k
D = Smallest prime in the form with n>=1, for b = 2 and odd k
E = Smallest n>=0 making the number prime, for b = 2
F = Smallest prime in the form with n>=0, for b = 2
G = Smallest n>=0 making the number prime, for b = 2 and odd k
H = Smallest prime in the form with n>=0, for b = 2 and odd k
I = Smallest n>=1 making the number prime for the reverse problem, for k = 2 and various bases b (only consider odd bases for the dual problems)
J = Smallest prime in the form with n>=1 for the reverse problem, for k = 2 and various bases b (only consider odd bases for the dual problems)

Type              A            B            C            D            E            F            G            H            I            J
Proth             A078680      A078683      A033809      xxxxxxx      A040076      A050921      A046067      A057025      A119624      xxxxxxx
Riesel            A050412      A052333      A108129      xxxxxxx      A040081      A038699      A046069      A057026      A119591      xxxxxxx
Dual Proth        xxxxxxx      xxxxxxx      A067760      A123252      xxxxxxx      xxxxxxx      xxxxxxx      xxxxxxx      A138066      A084713
Dual Riesel       xxxxxxx      xxxxxxx      A096502      A096822      xxxxxxx      xxxxxxx      xxxxxxx      xxxxxxx      A255707      A084714

Minimal primes (start with b+1) in base b=32 which are also Proth primes, Riesel primes, dual Proth primes, dual Riesel primes in base 2 (> 300 decimal digits):

Proth:

N0U(0^8362)1, which equals 11791*2^41816+1
NU(0^661863)1, which equals 383*2^3309321+1 (see http://www.prothsearch.com/riesel1a.html)
unsolved family 4{0}1, which corresponding to 2^n+1 with n == 2 mod 5 (n>2), first possible prime is the Fermat number F33, equivalent to this family searched to length (2^33-2)/5, see http://www.prothsearch.com/fermat.html
unsolved family G{0}1, which corresponding to 2^n+1 with n == 4 mod 5 (n>4), first possible prime is the Fermat number F34, equivalent to this family searched to length (2^34-4)/5, see http://www.prothsearch.com/fermat.html
unsolved family NG{0}1, which corresponding to 47*2^n+1 with n == 4 mod 5 (n>4), first such n is > 9000000, equivalent to this family searched to length 1800000, see http://www.prothsearch.com/riesel1.html
unsolved family UG{0}1, which corresponding to 61*2^n+1 with n == 4 mod 5 (n>4), first such n is > 3600000, equivalent to this family searched to length 720000, see http://www.prothsearch.com/riesel1.html

Riesel:

MS(V^415), which equals 733*32^415-1
unsolved family S{V}, which corresponding to 29*2^n-1 with n == 0 mod 5 (n>0), first such n is > 10000000, equivalent to this family searched to length 2000000, see http://www.noprimeleftbehind.net/cru...ers2.htm#R1024

Dual Proth:

G(0^264)K0F, which equals 2^1339+20495
8(0^1329)OV, which equals 2^6658+799
8(0^1716)AJ, which equals 2^8593+339
8(0^2217)AN, which equals 2^11098+343
2(0^5907)KT, which equals 2^29546+669
G(0^6654)F1, which equals 2^33284+481
G(0^7471)GF, which equals 2^37369+527
8(0^17186)MJ, which equals 2^85943+723
unsolved family 2{0}MD, which corresponding to 2^n+717 with n == 1 mod 5 (n>6)
unsolved family 4{0}1, which corresponding to 2^n+1 with n == 2 mod 5 (n>2), first possible prime is the Fermat number F33, equivalent to this family searched to length (2^33-2)/5, see http://www.prothsearch.com/fermat.html
unsolved family G{0}1, which corresponding to 2^n+1 with n == 4 mod 5 (n>4), first possible prime is the Fermat number F34, equivalent to this family searched to length (2^34-4)/5, see http://www.prothsearch.com/fermat.html

Dual Riesel:

(V^302)G3, which equals 2^1520-509
(V^387)C33, which equals 2^1950-20381
(V^478)8V, which equals 2^2400-737
(V^523)K9, which equals 2^2625-375
(V^2180)A3, which equals 2^10910-701
(V^16755)O3, which equals 2^83785-253
(V^17753)33, which equals 2^88775-925
unsolved family {V}KKV, which corresponding to 2^n-11617 with n == 0 mod 5 (n>10)
unsolved family {V}63, which corresponding to 2^n-829 with n == 0 mod 5 (n>5)
unsolved family {V}C9, which corresponding to 2^n-631 with n == 0 mod 5 (n>5)
unsolved family {V}3, which corresponding to 2^n-29 with n == 0 mod 5 (n>5)

Minimal primes (start with b+1) in base b=27 which are also Proth primes, Riesel primes, dual Proth primes, dual Riesel primes in base 3 (> 300 decimal digits):

Proth:

91(0^334)1, which equals 244*3^1005+1
BJ(0^383)1, which equals 316*3^1152+1
N3(0^401)1, which equals 208*3^1207+1
JD(0^7667)1, which equals 526*3^23004+1
PH(0^47890)1, which equals 692*3^143673+1
unsolved family 8JJ{0}1, which corresponding to 6364*3^n+1 with n == 0 mod 3 (n>0)

Riesel:

(none)

Dual Proth:

1(0^677)LD, which equals 3^2037+580
1(0^15935)HN, which equals 3^47811+482
unsolved family 1{0}JD, which corresponding to 3^n+526 with n == 0 mod 3 (n>3)

Dual Riesel:

(Q^221)LLLE, which equals 3^675-102208
(Q^223)LE, which equals 3^675-148
(Q^487)DJD, which equals 3^1470-9680
(Q^854)FFFA, which equals 3^2574-224846
(Q^7686)FA, which equals 3^23064-314

Minimal primes (start with b+1) in base b=25 which are also Proth primes, Riesel primes, dual Proth primes, dual Riesel primes in base 5 (> 300 decimal digits):

Proth:

70ED(0^253)1, which equals 109738*5^508+1
7J1J(0^254)1, which equals 121294*5^510+1
JD1J(0^262)1, which equals 305044*5^526+1
D701J(0^272)1, which equals 5187544*5^546+1
21D(0^277)1, which equals 1288*5^556+1
17K(0^299)1, which equals 164*5^601+1
12D(0^302)1, which equals 688*5^606+1
7D70D(0^343)1, which equals 2941888*5^688+1
1DJJ(0^354)1, which equals 24244*5^710+1
7D7D(0^432)1, which equals 117688*5^866+1
7DDJ(0^468)1, which equals 117844*5^938+1
1F(0^517)1, which equals 8*5^1037+1
K2(0^608)1, which equals 502*5^1218+1
11J7(0^915)1, which equals 16732*5^1832+1
78D(0^1128)1, which equals 4588*5^2258+1
D771(0^2113)1, which equals 207676*5^4228+1
1771(0^2858)1, which equals 20176*5^5718+1
77J7(0^3529)1, which equals 114232*5^7060+1
DJ7D(0^4962)1, which equals 215188*5^9926+1
ED7(0^7584)1, which equals 9082*5^15170+1
7ED(0^?)1, which equals 4738*5^?+1
1J71(0^96272)1, which equals 27676*5^192546+1 (see https://www.mersenneforum.org/showpo...3&postcount=18)
DKJ(0^246808)1, which equals
71JD(0^458549)1, which equals 110488*5^917100+1 (see http://www.primegrid.com/forum_thread.php?id=5087 and https://mersenneforum.org/showpost.p...25&postcount=3)
unsolved family D71J{0}1, which corresponding to 207544*5^n+1 with n == 0 mod 2 (n>0), first such n is > 700000, equivalent to this family searched to length 350000, see http://www.noprimeleftbehind.net/cru...25-reserve.htm

Riesel:

EFI(O^212), which equals 9144*5^424-1
3A(O^1029), which equals 86*5^2058-1
unsolved family EF{O}, which corresponding to 366*5^n-1 with n == 0 mod 2 (n>0), first such n is > 600000, equivalent to this family searched to length 300000, see http://www.noprimeleftbehind.net/cru...25-reserve.htm

Dual Proth:

Dual Riesel:

(O^223)359, which equals 5^452-13616
(O^509)2FI, which equals 5^1024-13982
(O^1039)E54, which equals 5^2084-6746
(O^10175)L8, which equals 5^20354-92

Minimal primes (start with b+1) in base b=36 which are also Proth primes, Riesel primes, dual Proth primes, dual Riesel primes in base 6 (> 300 decimal digits):

Proth:

(none)

Riesel:

P8(Z^390), which equals 909*6^780-1

Dual Proth:

(none)

Dual Riesel:

(Z^527)EX7, which equals 6^1060-27317
5(Z^2859)95, which equals 6^5723-967

[sweety439]

e.g. the minimal prime V(0^1328)444B in base 36, is the smallest prime in family V{0}444B in base 36

V{0}B --> always divisible by 7
V{0}4B --> always divisible by 31
V{0}44B --> always divisible by 5