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Old 2021-09-08, 05:55   #177
sweety439
 
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"99(4^34019)99 palind"
Nov 2016
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Some unsolved families are obvious through data for minimal primes (start with b+1) base b up to certain limit

Example 1: 80555551 is minimal prime (start with b+1) base b for b=10, thus these are unsolved families when searched to given limit if they possible contain primes:

* 80{5} searched to 5 5's (this family cannot have primes since all such numbers are divisible by 5)
* 8{5}1 searched to 5 5's (the smallest prime in this family is 8(5^20)1, but this prime is not minimal prime (start with b+1) for base b=10 since (5^11)1 is prime in this base and 20 >= 11)
* 0{5}1 searched to 5 5's (not considered, since this family has leading zeros)
* 8{5} searched to 5 5's (this family cannot have primes since all such numbers are divisible by 5)
* 0{5} searched to 5 5's (not considered, since this family has leading zeros)
* {5}1 searched to 5 5's (the smallest prime in this family is (5^11)1, and indeed minimal prime (start with b+1) for base b=10)
* {5} searched to 5 5's (this family cannot have primes since all such numbers are divisible by 5)

Example 2: 55555025 is minimal prime (start with b+1) base b for b=8, thus these are unsolved families when searched to given limit if they possible contain primes:

* {5}02 searched to 5 5's (this family cannot have primes since all such numbers are divisible by 2)
* {5}05 searched to 5 5's (this family cannot have primes since all such numbers are divisible by 5)
* {5}25 searched to 5 5's (the smallest prime in this family is (5^13)25, and indeed minimal prime (start with b+1) for base b=8)
* {5}0 searched to 5 5's (not considered, since this family has trailing zeros)
* {5}2 searched to 5 5's (this family cannot have primes since all such numbers are divisible by 2)
* {5}5 searched to 5 5's = {5} searched to 6 5's (this family cannot have primes since all such numbers are divisible by 5)

Example 3: 33333301 is minimal prime (start with b+1) base b for b=7, thus these are unsolved families when searched to given limit if they possible contain primes:

* {3}0 searched to 6 3's (not considered, since this family has trailing zeros)
* {3}1 searched to 6 3's (the smallest prime in this family is (3^16)1, and indeed minimal prime (start with b+1) for base b=7)
* {3} searched to 6 3's (this family cannot have primes since all such numbers are divisible by 3)

Example 4: 100000000000507 is minimal prime (start with b+1) base b for b=9, thus these are unsolved families when searched to given limit if they possible contain primes:

* 1{0}50 searched to 11 0's (not considered, since this family has trailing zeros)
* 1{0}57 searched to 11 0's (the smallest prime in this family is 1(0^25)57, and indeed minimal prime (start with b+1) for base b=9)
* 1{0}07 searched to 11 0's = 1{0}7 searched to 12 0's (this family cannot have primes since all such numbers are divisible by 2)
* 1{0}5 searched to 11 0's (this family cannot have primes since all such numbers are divisible by 2)
* 1{0}0 searched to 11 0's = 1{0} searched to 12 0's (not considered, since this family has trailing zeros)

Example 5: BBBBBB99B is minimal prime (start with b+1) base b for b=12, thus these are unsolved families when searched to given limit if they possible contain primes:

* {B}99 searched to 6 B's (this family cannot have primes since all such numbers are divisible by 3)
* {B}9B searched to 6 B's (this family cannot have primes since such numbers with even length are factored as difference of squares and such numbers with odd length are divisible by 13)
* {B}9 searched to 6 B's (this family cannot have primes since all such numbers are divisible by 3)
* {B}B searched to 6 B's = {B} searched to 7 B's (this family cannot have primes since all such numbers are divisible by 11)

Example 6: 500025 is minimal prime (start with b+1) base b for b=8, thus these are unsolved families when searched to given limit if they possible contain primes:

* 5{0}2 searched to 3 0's (this family cannot have primes since all such numbers are divisible by 2)
* 5{0}5 searched to 3 0's (this family cannot have primes since all such numbers are divisible by 5)
* {0}25 searched to 3 0's (not considered, since this family has leading zeros)
* 5{0} searched to 3 0's (not considered, since this family has trailing zeros)
* {0}2 searched to 3 0's (not considered, since this family has leading zeros)
* {0}5 searched to 3 0's (not considered, since this family has leading zeros)
* {0} searched to 3 0's (not considered, since this family has leading zeros and trailing zeros)

Example 7: 77774444441 is minimal prime (start with b+1) base b for b=8, thus these are unsolved families when searched to given limit if they possible contain primes:

* {7}1 searched to 4 7's
* {7}41 searched to 4 7's
* {7}441 searched to 4 7's
* {7}4441 searched to 4 7's
* {7}44441 searched to 4 7's
* {7}444441 searched to 4 7's
* {4}1 searched to 6 4's
* 7{4}1 searched to 6 4's
* 77{4}1 searched to 6 4's
* 777{4}1 searched to 6 4's

Example 8: 88888888833335 is minimal prime (start with b+1) base b for b=9, thus these are unsolved families when searched to given limit if they possible contain primes:

* {8}5 searched to 9 8's
* {8}35 searched to 9 8's
* {8}335 searched to 9 8's
* {8}3335 searched to 9 8's
* {3}5 searched to 4 3's
* 8{3}5 searched to 4 3's
* 88{3}5 searched to 4 3's
* 888{3}5 searched to 4 3's
* 8888{3}5 searched to 4 3's
* 88888{3}5 searched to 4 3's
* 888888{3}5 searched to 4 3's
* 8888888{3}5 searched to 4 3's
* 88888888{3}5 searched to 4 3's

Example 9: A44444777 is minimal prime (start with b+1) base b for b=11, thus these are unsolved families when searched to given limit if they possible contain primes:

* A{4} searched to 5 4's
* A{4}7 searched to 5 4's
* A{4}77 searched to 5 4's
* A{7} searched to 3 7's
* A4{7} searched to 3 7's
* A44{7} searched to 3 7's
* A444{7} searched to 3 7's
* A4444{7} searched to 3 7's

Example 10: 96664444 is minimal prime (start with b+1) base b for b=13, thus these are unsolved families when searched to given limit if they possible contain primes:

* 9{6} searched to 3 6's
* 9{6}4 searched to 3 6's
* 9{6}44 searched to 3 6's
* 9{6}444 searched to 3 6's
* 9{4} searched to 4 4's
* 96{4} searched to 4 4's
* 966{4} searched to 4 4's

Example 11: 88828823 is minimal prime (start with b+1) base b for b=11, thus these are unsolved families when searched to given limit if they possible contain primes:

* {8} searched to 5 8's
* {8}2 searched to 5 8's
* {8}3 searched to 5 8's
* 8882{8} searched to 5 8's
* {8}288 searched to 5 8's
* {8}23 searched to 5 8's
* 8882{8}2 searched to 5 8's
* 8882{8}3 searched to 5 8's
* {8}2882 searched to 5 8's
* {8}2883 searched to 5 8's

Example 12: B0BBB05BB is minimal prime (start with b+1) base b for b=13, thus these are unsolved families when searched to given limit if they possible contain primes:

* {B} searched to 6 B's
* B0{B} searched to 6 B's
* {B}0BB searched to 6 B's
* {B}5BB searched to 6 B's
* BBBB0{B} searched to 6 B's
* BBBB5{B} searched to 6 B's
* B0{B}0BB searched to 6 B's
* B0{B}5BB searched to 6 B's
* BBBB05{B} searched to 6 B's
* B0BBB0{B} searched to 6 B's
* B0BBB5{B} searched to 6 B's

Last fiddled with by sweety439 on 2021-10-16 at 13:18
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Old 2021-09-10, 05:37   #178
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Newest data (minimal primes (start with b+1) up to certain limit) for bases b=17 and b=18
Attached Files
File Type: txt minimal17.txt (61.4 KB, 39 views)
File Type: txt minimal18.txt (3.2 KB, 36 views)
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Old 2021-09-20, 03:10   #179
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"99(4^34019)99 palind"
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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
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Old 2021-09-20, 03:39   #180
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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.
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Old 2021-09-20, 15:52   #181
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"99(4^34019)99 palind"
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Quote:
Originally Posted by jwaltos View Post
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.
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File Type: txt current data.txt (272.6 KB, 33 views)
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Old 2021-09-20, 18:54   #182
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upload the text file for the proof for bases 2, 3, 4, 5, 6, 8, 10, 12
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File Type: txt proof.txt (52.0 KB, 32 views)
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Old 2021-09-21, 00:45   #183
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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
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Old 2021-09-22, 20:00   #184
sweety439
 
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"99(4^34019)99 palind"
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Quote:
Originally Posted by jwaltos View Post
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.
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Old 2021-09-22, 20:04   #185
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"99(4^34019)99 palind"
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Quote:
Originally Posted by jwaltos View Post
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.
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Old 2021-09-22, 23:28   #186
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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.
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Old 2021-10-02, 17:52   #187
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"99(4^34019)99 palind"
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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
File Type: txt 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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