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2021-09-10, 05:37   #178
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

3,049 Posts

Attached Files
 minimal17.txt (61.4 KB, 39 views) minimal18.txt (3.2 KB, 36 views)

 2021-09-20, 03:10 #179 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 3,049 Posts Newest condensed table for bases 2<=b<=16: Code: b number of minimal primes base b base-b form of largest known minimal prime base b length of largest known minimal prime base b algebraic ((a*bn+c)/d) form of largest known minimal prime base b 2 1 11 2 3 3 3 111 3 13 4 5 221 3 41 5 22 1(0^93)13 96 5^95+8 6 11 40041 5 5209 7 ≥71 (3^16)1 17 (7^17-5)/2 8 75 (4^220)7 221 (4*8^221+17)/7 9 ≥149 3(0^1158)11 1161 3*9^1160+10 10 77 5(0^28)27 31 5*10^30+27 11 ≥914 55(7^1011) 1013 (607*11^1011-7)/10 12 106 4(0^39)77 42 4*12^41+91 13 ≥2496 8(0^32017)111 32021 8*13^32020+183 14 ≥605 4(D^19698) 19699 5*14^19698-1 15 ≥1171 (7^155)97 157 (15^157+59)/2 16 ≥2050 D(B^32234) 32235 (206*16^32234-11)/15 Known values or lower bounds of the largest minimal prime base b for 2<=b<=36: Code: base (b) largest known minimal prime written in base b largest known minimal prime written in decimal length of largest known minimal prime written in base b 2 11 3 2 3 111 13 3 4 221 41 3 5 1(0^93)13 5^95+8 96 6 40041 5209 5 7 (3^16)1 (7^17-5)/2 17 8 (4^220)7 (4*8^221+17)/7 221 9 3(0^1158)11 3*9^1160+10 1161 10 5(0^28)27 5*10^30+27 31 11 55(7^1011) (607*11^1011-7)/10 1013 12 4(0^39)77 4*12^41+91 42 13 8(0^32017)111 8*13^32020+183 32021 14 4(D^19698) 5*14^19698-1 19699 15 (7^155)97 (15^157+59)/2 157 16 D(B^32234) (206*16^32234-11)/15 32235 17 F7(0^186767)1 262*17^186768+1 186770 18 8(0^298)B 8*18^299+11 300 19 FG(6^110984) (904*19^110984-1)/3 110986 20 C(D^2449) (241*20^2449-13)/19 2450 21 C(F^479147)0K (51*21^479149-1243)/4 479150 22 K(0^760)EC1 22^763*20+7041 764 23 9(E^800873) (106*23^800873-7)/11 800874 24 2(0^313)7 2*24^314+7 315 25 9(6^136965)M (37*25^136966+63)/4 136967 26 (M^8772)P (22*26^8773+53)/25 8773 27 A(0^109003)PM 10*27^109005+697 109006 28 O4(O^94535)9 (6092*28^94536-143)/9 94538 29 O(0^174236)FPL 24*29^174239+13361 174240 30 O(T^34205) 25*30^34205-1 34206 31 IE(L^29787) (5727*31^29787-7)/10 29789 32 S(U^9748)L (898*32^9749-309)/31 9750 33 N7(0^610411)1 766*33^610412+1 610414 34 US(0^9374)R 1048*34^9375+27 9377 35 1B(0^56061)1 46*35^56062+1 56064 36 (P^81993)SZ (5*36^81995+821)/7 81995
 2021-09-20, 03:39 #180 jwaltos     Apr 2012 Gracie on alert. 1100100012 Posts Sweety, could you please explain what it is you're doing? More specifically, what is your end objective, goal or result? You have a lot of "stuff" but how does it support whatever it is you are doing? Second, by posting, what is it that you hope others, like myself, will gain by reading it all? Cheers.
2021-09-20, 15:52   #181
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

3,049 Posts

Quote:
 Originally Posted by jwaltos Sweety, could you please explain what it is you're doing? More specifically, what is your end objective, goal or result? You have a lot of "stuff" but how does it support whatever it is you are doing? Second, by posting, what is it that you hope others, like myself, will gain by reading it all? Cheers.
I have a project: To find and proof all minimal primes (start with b+1) in bases 2<=b<=36, this text file is the current data, only bases 2, 3, 4, 5, 6, 8, 10, 12 were solved.

You can see my article.
Attached Files
 current data.txt (272.6 KB, 33 views)

2021-09-20, 18:54   #182
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

3,049 Posts

upload the text file for the proof for bases 2, 3, 4, 5, 6, 8, 10, 12
Attached Files
 proof.txt (52.0 KB, 32 views)

 2021-09-21, 00:45 #183 jwaltos     Apr 2012 Gracie on alert. 40110 Posts Thanks! I took a look at your article and the two associated files. That's a lot of number crunching. Have you made any conjectures, developed any distributions that may follow a pattern (usually tough to discern) or have been able to consolidate your observations within specific boundary conditions? How long have you been working on your project? I started one back in 2001 anticipating that it would be done in three years...20 years later I'm still working on it. Last fiddled with by jwaltos on 2021-09-21 at 00:47
2021-09-22, 20:00   #184
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

1011111010012 Posts

Quote:
 Originally Posted by jwaltos Thanks! I took a look at your article and the two associated files. That's a lot of number crunching. Have you made any conjectures, developed any distributions that may follow a pattern (usually tough to discern) or have been able to consolidate your observations within specific boundary conditions? How long have you been working on your project? I started one back in 2001 anticipating that it would be done in three years...20 years later I'm still working on it.
I have a conjecture in my article: every x{y}z (i.e. xyyy...yyyz) simple family in every base b (the algebraic form is (a*b^n+c)/gcd(a+c,b-1) with fixed integers a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1) which can be proven to contain no primes > base (by covering congruence, by algebra factorization, or by the combine of them) contains a prime > base, in fact, I conjectured that every x{y}z (i.e. xyyy...yyyz) simple family in every base b (the algebraic form is (a*b^n+c)/gcd(a+c,b-1) with fixed integers a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1) which can be proven to contain no primes > base or only contain finite primes > base (by covering congruence, by algebra factorization, or by the combine of them) contains infinitely many primes > base.

2021-09-22, 20:04   #185
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

3,049 Posts

Quote:
 Originally Posted by jwaltos Thanks! I took a look at your article and the two associated files. That's a lot of number crunching. Have you made any conjectures, developed any distributions that may follow a pattern (usually tough to discern) or have been able to consolidate your observations within specific boundary conditions? How long have you been working on your project? I started one back in 2001 anticipating that it would be done in three years...20 years later I'm still working on it.
You can help me to completely solve bases 7, 9, and 11 through 36, I think that my data for base 7 is complete, but I cannot prove this.

 2021-09-22, 23:28 #186 jwaltos     Apr 2012 Gracie on alert. 401 Posts You were kind enough to respond so I'll try to help out. I can't predict when I'll have something but when I do I'll send it via PM rather than posting within the forum.
2021-10-02, 17:52   #187
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

3,049 Posts

PARI/GP program of this problem attached (not completely done, continue updating ....)

Note: in this program, vector [a,b,[c],d,e] means simple family ab{c}de = abccc...cccde, and vector [a,b,[c,d],e,f] means nonsimple family ab{c,d}ef, and all numbers in these vectors are the digit value of the base b digits, i.e. 10 for A, 11 for B, 12 for C, ...
Attached Files
 program of minimal primes.txt (2.3 KB, 14 views)

Last fiddled with by sweety439 on 2021-10-02 at 19:37

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