- **sweety439**
(*https://www.mersenneforum.org/forumdisplay.php?f=137*)

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

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 71JD(0^458549)1, which equals 110488*5^917100+1 (see [URL="http://www.primegrid.com/forum_thread.php?id=5087"]http://www.primegrid.com/forum_thread.php?id=5087[/URL] and [URL="https://mersenneforum.org/showpost.php?p=334925&postcount=3"]https://mersenneforum.org/showpost.php?p=334925&postcount=3[/URL]) 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 |

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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