20221201, 15:39  #386 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×7×263 Posts 
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 > n1 (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

20221208, 08:26  #387 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·7·263 Posts 
(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 differenceofsquares 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 differenceofsquares factorization) 
20221220, 17:34  #388 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
E62_{16} Posts 
Newest condensed table of the problem: (no data available for bases 29, 31, 33, 35)
Code:
b number of quasiminimal primes base b baseb form of largest known quasiminimal prime base b length of largest known quasiminimal prime base b length of largest known quasiminimal prime base b when written in decimal algebraic ((a×bn+c)/d) form of largest known quasiminimal 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)) Last fiddled with by sweety439 on 20221220 at 17:44 
20221220, 17:43  #389 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
E62_{16} Posts 
Proth primes base b: Primes of the form k*b^n+1
Riesel primes base b: Primes of the form k*b^n1 Dual Proth primes base b: Primes of the form b^n+k Dual Riesel primes base b: Primes of the form b^nk 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 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 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 (2^332)/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 (2^344)/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*32^4151 unsolved family S{V}, which corresponding to 29*2^n1 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 (2^332)/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 (2^344)/5, see http://www.prothsearch.com/fermat.html Dual Riesel: (V^302)G3, which equals 2^1520509 (V^387)C33, which equals 2^195020381 (V^478)8V, which equals 2^2400737 (V^523)K9, which equals 2^2625375 (V^2180)A3, which equals 2^10910701 (V^16755)O3, which equals 2^83785253 (V^17753)33, which equals 2^88775925 unsolved family {V}KKV, which corresponding to 2^n11617 with n == 0 mod 5 (n>10) unsolved family {V}63, which corresponding to 2^n829 with n == 0 mod 5 (n>5) unsolved family {V}C9, which corresponding to 2^n631 with n == 0 mod 5 (n>5) unsolved family {V}3, which corresponding to 2^n29 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 3^675102208 (Q^223)LE, which equals 3^675148 (Q^487)DJD, which equals 3^14709680 (Q^854)FFFA, which equals 3^2574224846 (Q^7686)FA, which equals 3^23064314 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...25reserve.htm Riesel: EFI(O^212), which equals 9144*5^4241 3A(O^1029), which equals 86*5^20581 unsolved family EF{O}, which corresponding to 366*5^n1 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...25reserve.htm Dual Proth: Dual Riesel: (O^223)359, which equals 5^45213616 (O^509)2FI, which equals 5^102413982 (O^1039)E54, which equals 5^20846746 (O^10175)L8, which equals 5^2035492 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*6^7801 Dual Proth: (none) Dual Riesel: (Z^527)EX7, which equals 6^106027317 5(Z^2859)95, which equals 6^5723967 Last fiddled with by sweety439 on 20221231 at 01:48 
20221224, 00:37  #390 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×7×263 Posts 
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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