Minimal set of the strings for primes with at least two digits

[sweety439]

Update the text file for solved minimal prime (start with b+1) set

(Note: the set of base 7 is only conjectured, I think this set is complete but I cannot prove it, and I tried to find more primes in this set (find simple families which have no primes in current set but not ruled out as only contain composites (only count numbers > base)) and with no success, thus I think that this set is complete)

[sweety439]

Update the file of the condensed table (the current status for bases 2<=b<=16)

[VBCurtis]

Your work does not rate daily posts to update us.

If you continue to post to this thread every single day, you're going to find yourself with time off again.

Try monthly update posts. Yes, monthly.

You can edit your previously posted attachments without triggering a new-post notice to all the mods- try that too. But if you keep drawing attention to your endless procession of trivial update posts, you're likely to lose the ability to make those posts.

[sweety439]

The largest possible appearance for given digit d in minimal prime (start with b+1) in base b: (note: the case for b=7 and b=9 are only conjectured (assume my sets for these bases are complete), not proven)

If base b has repunit primes, then the largest possible appearance for digit d=1 in minimal prime (start with b+1) in base b is the length of smallest repunit prime base b (i.e. A084740(b)), the first bases which do not have repunit primes are 9, 25, 32, 49, 64, ...

b=2, d=0: 0
b=2, d=1: 2 (the prime 11)
b=3, d=0: 0
b=3, d=1: 3 (the prime 111)
b=3, d=2: 1 (the primes 12 and 21)
b=4, d=0: 0
b=4, d=1: 2 (the prime 11)
b=4, d=2: 2 (the prime 221)
b=4, d=3: 1 (the primes 13, 23, 31)
b=5, d=0: 93 (the prime 109313)
b=5, d=1: 3 (the prime 111)
b=5, d=2: 1 (the primes 12, 21, 23, 32)
b=5, d=3: 4 (the prime 33331)
b=5, d=4: 4 (the primes 14444 and 44441)
b=6, d=0: 2 (the prime 40041)
b=6, d=1: 2 (the prime 11)
b=6, d=2: 1 (the primes 21 and 25)
b=6, d=3: 1 (the primes 31 and 35)
b=6, d=4: 3 (the prime 4441)
b=6, d=5: 1 (the primes 15, 25, 35, 45, 51)
b=7, d=0: 7 (the prime 5100000001)
b=7, d=1: 5 (the prime 11111)
b=7, d=2: 3 (the prime 1222)
b=7, d=3: 16 (the prime 3161)
b=7, d=4: 2 (the primes 344, 445, 544, 4504, 40054)
b=7, d=5: 4 (the prime 35555)
b=7, d=6: 2 (the prime 6634)
b=8, d=0: 3 (the prime 500025)
b=8, d=1: 3 (the prime 111)
b=8, d=2: 2 (the prime 225)
b=8, d=3: 3 (the prime 3331)
b=8, d=4: 220 (the prime 42207)
b=8, d=5: 14 (the prime 51325)
b=8, d=6: 2 (the primes 661 and 667)
b=8, d=7: 12 (the prime 7121)
b=9, d=0: 1158 (the prime 30115811)
b=9, d=1: 36 (the prime 56136)
b=9, d=2: 4 (the prime 22227)
b=9, d=3: 8 (the prime 8333333335)
b=9, d=4: 11 (the prime 5411)
b=9, d=5: 4 (the prime 55551)
b=9, d=6: 329 (the prime 763292)
b=9, d=7: 687 (the prime 2768607)
b=9, d=8: 19 (the prime 819335)
b=10, d=0: 28 (the prime 502827)
b=10, d=1: 2 (the prime 11)
b=10, d=2: 3 (the prime 2221)
b=10, d=3: 1 (the primes 13, 23, 31, 37, 43, 53, 73, 83, 349)
b=10, d=4: 2 (the prime 449)
b=10, d=5: 11 (the prime 5111)
b=10, d=6: 4 (the prime 666649)
b=10, d=7: 2 (the primes 277, 577, 727, 757, 787, 877)
b=10, d=8: 2 (the prime 881)
b=10, d=9: 3 (the prime 9949)
b=12, d=0: 39 (the prime 403977)
b=12, d=1: 2 (the prime 11)
b=12, d=2: 3 (the prime 222B)
b=12, d=3: 1 (the primes 31, 35, 37, 3B)
b=12, d=4: 3 (the prime 4441)
b=12, d=5: 2 (the primes 565 and 655)
b=12, d=6: 2 (the prime 665)
b=12, d=7: 3 (the primes 4777 and 9777)
b=12, d=8: 1 (the primes 81, 85, 87, 8B)
b=12, d=9: 4 (the prime 9999B)
b=12, d=A: 4 (the prime AAAA1)
b=12, d=B: 7 (the prime BBBBBB99B)

[sweety439]

Update the text file for all known minimal primes (start with b+1) in bases 2<=b<=16

Note: Only bases 2, 3, 4, 5, 6, 8, 10, 12 are completely solved.

[sweety439]

New minimal prime (start with b+1) in base b is found for b=650: 3:{649}^(498101), see https://mersenneforum.org/showpost.p...&postcount=931

Added it to excel file https://docs.google.com/spreadsheets...RwmKME/pubhtml

Base 108 is an interesting base since ....

* For the family {1}, length 2 is prime, but the next prime is large (length 449)
* For the family 1{0}1, length 2 is prime, the next prime is not known
* For the family y{z}, first prime is large (length 411)
* For the family 11{0}1, first prime is large (length 400)
* For the family {y}z, first prime is large (length 492) (note that length 1 is also prime, but length 1 is not allowed in this project)
* For the family 6{0}1, first prime is large (length 16318)
* For the family #{z} (# = (base/2)-1)), first prime is large (length 7638)

This situation is not common in bases with many divisors, but although 108 has many divisors, this situation occurs in this base, this is why this base is interesting :))

Also base 282 ....

* For the family A{0}1, first prime is large (length 1474)
* For the family C{0}1, first prime is large (length 2957)
* For the family z{0}1, first prime is large (length 277)

http://www.noprimeleftbehind.net/cru...82-reserve.htm only tells you that all these three families have a prime with length <= 100001 ....

* For the family 7{z}, first prime is large (length 21413)
* For the family 10{z}, first prime is large (length 780)

http://www.noprimeleftbehind.net/cru...82-reserve.htm only tells you that the farmer family has a prime with length <= 100001, and the letter family has a prime with length <= 100002

[sweety439]

Searched 1{0}2 (b^n+2) and {z}y (b^n-2) (for bases 2<=b<=1024) up to n=5000 and found the (probable) primes 485^3164-2 and 487^3775-2

b^n+2 for all remain bases b<=711 and b^n-2 for all remain bases b<=533 are checked to n=5000 with no (probable) primes found.

I will reserve 1{0}z (b^n+(b-1)) and {z}1 (b^n-(b-1)) (for bases 2<=b<=1024) (also up to n=5000) after this reservation was done.

These families were already tested to large n: (only consider families which must be minimal primes (start with b+1))

{1}: https://web.archive.org/web/20021111...ds/primes.html https://raw.githubusercontent.com/xa...iesel%20k1.txt

1{0}1: http://jeppesn.dk/generalized-fermat.html http://www.noprimeleftbehind.net/crus/GFN-primes.htm

2{0}1, 3{0}1, 4{0}1, 5{0}1, 6{0}1, 7{0}1, 8{0}1, 9{0}1, A{0}1, B{0}1, C{0}1: https://www.rieselprime.de/ziki/Prot..._bases_least_n

1{z}, 2{z}, 3{z}, 4{z}, 5{z}, 6{z}, 7{z}, 8{z}, 9{z}, A{z}, B{z}: https://www.rieselprime.de/ziki/Ries..._bases_least_n

z{0}1: https://www.rieselprime.de/ziki/Williams_prime_MP_least

y{z}: https://harvey563.tripod.com/wills.txt https://www.rieselprime.de/ziki/Williams_prime_MM_least

and this table was updated

[sweety439]

Records of length in these families for various bases: (format: base (length)) (bases with "NB" or "RC" for given family are not counted)

1{0}1:

2 (2)
14 (3)
34 (5)
38 (>8388608)

1{0}2:

3 (2)
23 (12)
47 (114)
89 (256)
167 (>100001)

1{0}3:

4 (2)
22 (3)
32 (4)
46 (21)
292 (40)
382 (256)
530 (1399)
646 (>5000)

1{0}4:

5 (3)
23 (7)
53 (13403)
139 (>25000)

1{0}z:

2 (2)
5 (3)
14 (17)
32 (109)
80 (195)
107 (1401)
113 (20089)
123 (64371)
173? (>5000)

{1}:

2 (2)
3 (3)
7 (5)
11 (17)
19 (19)
35 (313)
39 (349)
51 (4229)
91 (4421)
152 (270217)
185? (>66337)

1{2}:

3 (2)
7 (4)
31 (76)
97 (1128)
265 (2301)
355 (>5000)

1{3}:

4 (2)
5 (3)
17 (5)
29 (19)
46 (82)
59 (85)
71 (197)
107 (>5000)

1{4}:

5 (5)
11 (19)
17 (61)
83 (>5000)

1{z}:

2 (2)
5 (5)
20 (11)
29 (137)
67 (769)
107 (21911)
170 (166429)
581 (>400001)

2{0}1:

3 (2)
12 (4)
17 (48)
38 (2730)
101 (192276)
218 (333926)
365? (>300001)

2{0}3:

4 (2)
23 (3)
44 (4)
58 (6)
59 (18)
79 (>5000)

[sweety439]

all b^n+-2 and b^n+-(b-1) for 2<=b<=1024 tested to n=5000

status files attached

(z means b-1, y means b-2)

b^n+2 = (1000...0002) base b = 10002
b^n-2 = (zzz...zzzy) base b = zzzzy
b^n+(b-1) = (1000...000z) base b = 1000z
b^n-(b-1) = (zzz...zzz1) base b = zzzz1

edit: ((b-2)*b^n+1)/(b-1) (family yyyyz) also tested to n=5000

[sweety439]

There are many conjectures related to this project (find all minimal primes (start with b+1) in bases 2<=b<=1024):

* Are there infinitely many Mersenne primes? (related to family {1} in base 2)
* Are there infinitely many Fermat primes? (related to family 1{0}1 in base 2)
* Are there infinitely many Wagstaff primes? (related to family {2}3 in base 4)
* Are there infinitely many repunit primes? (related to family {1} in base 10)
* Are there infinitely many generalized Fermat primes base 10? (related to family 1{0}1 in base 10)
* Odd perfect numbers search (related to family {1} in prime bases)
* n-hyperperfect numbers search (related to family {z}1 in base n+1 if n+1 is prime)
* Are there infinitely many triples of 3 consecutive numbers with all have primitive roots? (related to families {1}, {2}1, {1}2, 1{0}2, 1{2}, 2{0}1 in base 3)
* New Mersenne conjecture (primes p in families {1} in base 2, 1{0}1 in base 2, {3}1 in base 4, 1{0}3 in base 4, and related to Mersenne primes ({1} in base 2) and Wagstaff primes ({2}3 in base 4))
* "Dividing Phi" category (related to family 2{0}1 in bases == 11 mod 12)
* Sierpinski problem (related to family *{0}1 in base 2)
* Riesel problem (related to family *{1} in base 2)
* Dual Sierpinski problem (related to family 1{0}* in base 2)
* Dual Riesel problem (related to family {1}* in base 2)
* Generalized Sierpinski problem base b (related to family *{0}1 in base b)
* Generalized Riesel problem base b (related to family *{z} in base b)
* Problem 197 (related to family *{1} in base 10)

Also related types of primes (left-truncatable primes, right-truncatable primes, two-sides primes, detelable primes, permutable primes, circular primes, palindromic primes, etc.)

This project specially to prime bases are more related to the conjectures because....

* {1} family, related to odd perfect numbers search (factorization of numbers in this family)
* 2{0}1 family, related to "Dividing Phi" category (when the prime is == 11 mod 12) (and when the prime is == 5 mod 12, it will divide Phi(2*p^n,2) instead of Phi(p^n,2))
* z{0}1 family, related to inverse of Euler totient function (it is usually the case that, for prime p and k > 1, the first time the totient function phi(n) has value p^k - p^(k-1) is for n = p^k. However, this is not true when p^k - p^(k-1) + 1 is prime)
* {z}1 family, related to (this prime minus 1)-hyperperfect numbers search

also, the ring of the b-adic numbers (related to base b numbers, see https://en.wikipedia.org/wiki/Automorphic_number) is a field if and only if b is prime or prime power

[sweety439]

We can use the sense of http://www.iakovlev.org/zip/riesel2.pdf to conclude that the unsolved families (unsolved families are families which are neither primes (>base) found nor can be proven to contain no primes > base) eventually should yield a prime, e.g. for the base 11 unsolved family 5{7}:

5(7^n) = (57*11^n-7)/10, but there is no n satisfying that 57*11^n and 7 are both r-th powers for some r>1 (since 7 is not perfect power), nor there is n satisfying that 57*11^n and -7 are (one is 4th power, another is of the form 4*m^4) (since -7 is neither 4th power nor of the form 4*m^4), thus, 5(7^n) has no algebra factors for any n, thus 5(7^n) eventually should yield a prime unless it can be proven to contain no primes > base using covering congruence, and we have:

5(7^n) is divisible by 2 for n == 1 mod 2
5(7^n) is divisible by 13 for n == 2 mod 12
5(7^n) is divisible by 17 for n == 4 mod 16
5(7^n) is divisible by 5 for n == 0 mod 5
5(7^n) is divisible by 23 for n == 6 mod 22
5(7^n) is divisible by 601 for n == 8 mod 600
5(7^n) is divisible by 97 for n == 12 mod 48
...

and it does not appear to be any covering set of primes, so there must be a prime at some point.

(see post #103 for examples of families which can be proven to contain no primes > base)