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-   -   Minimal set of the strings for primes with at least two digits (https://www.mersenneforum.org/showthread.php?t=24972)

 sweety439 2022-12-20 17:43

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)

[CODE]
Type A B C D E F G H I J
Proth [URL="https://oeis.org/A078680"]A078680[/URL] [URL="https://oeis.org/A078683"]A078683[/URL] [URL="https://oeis.org/A033809"]A033809[/URL] xxxxxxx [URL="https://oeis.org/A040076"]A040076[/URL] [URL="https://oeis.org/A050921"]A050921[/URL] [URL="https://oeis.org/A046067"]A046067[/URL] [URL="https://oeis.org/A057025"]A057025[/URL] [URL="https://oeis.org/A119624"]A119624[/URL] xxxxxxx
Riesel [URL="https://oeis.org/A050412"]A050412[/URL] [URL="https://oeis.org/A052333"]A052333[/URL] [URL="https://oeis.org/A108129"]A108129[/URL] xxxxxxx [URL="https://oeis.org/A040081"]A040081[/URL] [URL="https://oeis.org/A038699"]A038699[/URL] [URL="https://oeis.org/A046069"]A046069[/URL] [URL="https://oeis.org/A057026"]A057026[/URL] [URL="https://oeis.org/A119591"]A119591[/URL] xxxxxxx
Dual Proth xxxxxxx xxxxxxx [URL="https://oeis.org/A067760"]A067760[/URL] [URL="https://oeis.org/A123252"]A123252[/URL] xxxxxxx xxxxxxx xxxxxxx xxxxxxx [URL="https://oeis.org/A138066"]A138066[/URL] [URL="https://oeis.org/A084713"]A084713[/URL]
Dual Riesel xxxxxxx xxxxxxx [URL="https://oeis.org/A096502"]A096502[/URL] [URL="https://oeis.org/A096822"]A096822[/URL] xxxxxxx xxxxxxx xxxxxxx xxxxxxx [URL="https://oeis.org/A255707"]A255707[/URL] [URL="https://oeis.org/A084714"]A084714[/URL]
[/CODE]

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 [URL="http://www.prothsearch.com/riesel1a.html"]http://www.prothsearch.com/riesel1a.html[/URL])
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 [URL="http://www.prothsearch.com/fermat.html"]http://www.prothsearch.com/fermat.html[/URL]
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 [URL="http://www.prothsearch.com/fermat.html"]http://www.prothsearch.com/fermat.html[/URL]
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 [URL="http://www.prothsearch.com/riesel1.html"]http://www.prothsearch.com/riesel1.html[/URL]
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 [URL="http://www.prothsearch.com/riesel1.html"]http://www.prothsearch.com/riesel1.html[/URL]

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 [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm#R1024"]http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm#R1024[/URL]

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 [URL="http://www.prothsearch.com/fermat.html"]http://www.prothsearch.com/fermat.html[/URL]
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 [URL="http://www.prothsearch.com/fermat.html"]http://www.prothsearch.com/fermat.html[/URL]

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 [URL="https://www.mersenneforum.org/showpost.php?p=94583&postcount=18"]https://www.mersenneforum.org/showpost.php?p=94583&postcount=18[/URL])
DKJ(0^246808)1, which equals
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 [URL="http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base25-reserve.htm"]http://www.noprimeleftbehind.net/crus/Sierp-conjecture-base25-reserve.htm[/URL]

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 [URL="http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base25-reserve.htm"]http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base25-reserve.htm[/URL]

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 2022-12-24 00:37

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

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