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[ May 6, 2022 ] [ Last update May 27, 2024 ]
Minimal primes in other bases
A hard problem by Xinyao Chen (email)

A string a is a subsequence of another string b, if a can be
obtained from b by deleting zero or more of the characters in b.
For example, 514 is a substring of 251664.
The empty string is a subsequence of every string.

Two strings a and b are comparable if either a is a substring of b,
or b is a substring of a.

A surprising result from formal language theory is that every set
of pairwise incomparable strings is finite. This means that from
any set of strings we can find its minimal elements.

A string a in a set of strings S is minimal if whenever b
(an element of S) is a substring of a, we have b = a.

This set must be finite! See

https://primes.utm.edu/glossary/xpage/MinimalPrime.html

For example, if our set is the set of prime numbers (written in radix 10),
then we get the set {2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949,
9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049},
(A071062)

and if our set is the set of composite numbers (written in radix 10),
then we get the set {4, 6, 8, 9, 10, 12, 15, 20, 21, 22, 25, 27, 30, 32, 33, 35,
50, 51, 52, 55, 57, 70, 72, 75, 77, 111, 117, 171, 371, 711, 713, 731}
(A071070)

and if our set is the set of powers of 2 (written in radix 10), then we get the set
{1, 2, 4, 8, 65536}
(A071071)
(This set is conjectured to be complete, but not proven, and is
complete if all powers of 2 which are > 65536 contain at least
one of the digits {1, 2, 4, 8}, and this is extremely likely,
since it is conjectured that 2^n contain at least one digit 1 if
n > 91, 2^n contain at least one digit 2 if n > 168, 2^n contain
at least one digit 4 if n > 107, 2^n contain at least one digit 8 if
n > 78, see a071071.pdf for more information)

and if our set is the set of squares (written in radix 10), then we get the set
{1, 4, 9, 25, 36, 576, 676, 7056, 80656, 665856, 2027776, 2802276, 22282727076,
77770707876, 78807087076, 7888885568656, 8782782707776, 72822772707876,
555006880085056, 782280288087076, 827702888070276, 888288787822276,
2282820800707876, 7880082008070276, 80077778877070276, 88778000807227876,
782828878078078276, 872727072820287876, 2707700770820007076,
7078287780880770276, 7808287827720727876, 8008002202002207876,
27282772777702807876, 70880800720008787876, 72887222220777087876,
80028077888770207876, 80880700827207270276, 87078270070088278276,
88002002000028027076, 2882278278888228807876, 8770777780888228887076,
77700027222828822007876, 702087807788807888287876,
788708087882007280808827876, 880070008077808877000002276,
888000227087070707880827076, 888077027227228277087787076,
888588886555505085888555556, 7770000800780088788282227776,
7782727788888878708800870276, 5000060065066660656065066555556,
8070008800822880080708800087876, 80787870808888808272077777227076,
800008088070820870870077778827876, 822822722220080888878078820887876, ...}
(A130448)
(This set is currently not known, and might be extremely difficult to find, but by the
theorem above, this set must be finite, it is known that no repdigits (numbers whose all
digits are same) are squares, and for some families like {7}6 we can prove that they
contain no squares, by quadratic residue
(since all numbers in the family {7}6 are == 6 mod 7 and == 6, 10 mod 11,
but squares are == 0, 1, 2, 4 mod 7 and == 0, 1, 3, 4, 5, 9 mod 11), but we still cannot
prove that the family {5}6 contains no squares, since {5}6 can be
quadratic residue mod first few primes simultaneously, and (5^^n)6 == 4 mod 8 if
n ⩾ 2 (thus, (5^^n)6 is quadratic residue mod 2^r for all r if n ⩾ 2), e.g.
(5^^11)6 is quadratic residue mod all primes < 29 (in fact, mod all numbers < 29),
(5^^389)6 is quadratic residue mod all primes < 67 (in fact, mod all numbers < 67),
(5^^27719)6 is quadratic residue mod all primes < 97 (in fact, mod all numbers < 97), and
(5^^196559)6 is quadratic residue mod all primes < 103 (in fact, mod all numbers < 103),
maybe a theorem like Catalan's conjecture (i.e. b^r+-1 cannot be square if r > 1
unless b = 2 and r = 3) and Ramanujan–Nagell equation (i.e. 2^r-7 cannot be square
if r > 15) also exists for the family {5}6
(the smallest such square (if exists) must be in this set)?
e.g. the element 5000060065066660656065066555556 in the set of squares
is the smallest square containing only the digits 0, 5, 6,
see Squares containing at most three distinct digits and
Perfect squares consisting of only 3 different digits ,
and we still cannot prove that no squares (besides 676) contain no digits
other than {6, 7, 8} (see Squares containing at most three distinct digits and
Perfect squares consisting of only 3 different digits, the smallest
such square (if exists) must be in this set if it does not contain 676 as substring.
Since all squares containing no digits other than {6, 7, 8} end with 76
(since 66, 67, 68, 77, 78, 86, 87, 88 cannot be the final two digits of a square),
thus a square containing no digits other than {6, 7, 8} does not contain 676 as
substring if and only if it is of the form {7,8}*76 (using the notation in Shallit's
article), thus the smallest square in the family {7,8}*76 must be in this set,
and we cannot find a square in this family nor can prove that this family
contains no squares).

Besides, if our set is the set of prime numbers written in radix b, then we get these sets:

bwe get the set
2{10, 11}
3{2, 10, 111}
4{2, 3, 11}
5{2, 3, 10, 111, 401, 414, 14444, 44441}
6{2, 3, 5, 11, 4401, 4441, 40041}
7{2, 3, 5, 10, 14, 16, 41, 61, 11111}

These are already researched in
Minimal Elements for the Prime Numbers (bases 2 ⩽ b ⩽ 30),
Minimal Slides for the Prime Numbers (bases 2 ⩽ b ⩽ 30) and
The Kernel of the Primes for Bases Two through Ten (bases 2 ⩽ b ⩽ 10).
For the data of these sets and the unsolved families see
https://github.com/curtisbright/mepn-data/tree/master/data (bases 2 ⩽ b ⩽ 30)
and https://github.com/RaymondDevillers/primes (bases 28 ⩽ b ⩽ 50)
using A–Z to represent digit values 10 to 35, and using a–n to represent digit values 36 to 49.

Now, let's consider: if our set is the set of prime numbers   > b 
written in radix b, then we get the sets:

(using A−F to represent digit values 10 to 15)

bthe set for base b
2{11}
3{12,
21,
111}
4{11,
13,
23,
31,
221}
5
{12,
21,
23,
32,
34,
43,
104,
111,
131,
133,
313,
401,
414,
3101,
10103,
14444,
30301,
33001,
33331,
44441,
300031,
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013}
6
{11,
15,
21,
25,
31,
35,
45,
51,
4401,
4441,
40041}
7
{14,
16,
23,
25,
32,
41,
43,
52,
56,
61,
65,
113,
115,
131,
133,
155,
212,
221,
304,
313,
335,
344,
346,
364,
445,
515,
533,
535,
544,
551,
553,
1022,
1051,
1112,
1202,
1211,
1222,
2111,
3031,
3055,
3334,
3503,
3505,
3545,
4504,
4555,
5011,
5455,
5545,
5554,
6034,
6634,
11111,
11201,
30011,
30101,
31001,
31111,
33001,
33311,
35555,
40054,
100121,
150001,
300053,
351101,
531101,
1100021,
33333301,
5100000001,
33333333333333331}
8
{13,
15,
21,
23,
27,
35,
37,
45,
51,
53,
57,
65,
73,
75,
107,
111,
117,
141,
147,
161,
177,
225,
255,
301,
343,
361,
401,
407,
417,
431,
433,
463,
467,
471,
631,
643,
661,
667,
701,
711,
717,
747,
767,
3331,
3411,
4043,
4443,
4611,
5205,
6007,
6101,
6441,
6477,
6707,
6777,
7461,
7641,
47777,
60171,
60411,
60741,
444641,
500025,
505525,
3344441,
4444477,
5500525,
5550525,
55555025,
444444441,
744444441,
77774444441,
7777777777771,
555555555555525,
44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447}
9
{12,
14,
18,
21,
25,
32,
34,
41,
45,
47,
52,
58,
65,
67,
74,
78,
81,
87,
117,
131,
135,
151,
155,
175,
177,
238,
272,
308,
315,
331,
337,
355,
371,
375,
377,
438,
504,
515,
517,
531,
537,
557,
564,
601,
638,
661,
702,
711,
722,
735,
737,
751,
755,
757,
771,
805,
838,
1011,
1015,
1101,
1701,
2027,
2207,
3017,
3057,
3101,
3501,
3561,
3611,
3688,
3868,
5035,
5051,
5071,
5101,
5501,
5554,
5705,
5707,
7017,
7075,
7105,
7301,
8535,
8544,
8555,
8854,
20777,
22227,
22777,
30161,
33388,
50161,
50611,
53335,
55111,
55535,
55551,
57061,
57775,
70631,
71007,
77207,
100037,
100071,
100761,
105007,
270707,
301111,
305111,
333035,
333385,
333835,
338885,
350007,
500075,
530005,
555611,
631111,
720707,
2770007,
3030335,
7776662,
30300005,
30333335,
38333335,
51116111,
70000361,
300030005,
300033305,
351111111,
1300000007,
5161111111,
8333333335,
300000000035,
311111111161,
544444444444,
2000000000007,
5700000000001,
7270000000007,
88888888833335,
100000000000507,
5111111111111161,
7277777777777777707,
8888888888888888888335,
30000000000000000000051,
1000000000000000000000000057,
56111111111111111111111111111111111111,
7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662,
27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707,
300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011}
10
{11,
13,
17,
19,
23,
29,
31,
37,
41,
43,
47,
53,
59,
61,
67,
71,
73,
79,
83,
89,
97,
227,
251,
257,
277,
281,
349,
409,
449,
499,
521,
557,
577,
587,
727,
757,
787,
821,
827,
857,
877,
881,
887,
991,
2087,
2221,
5051,
5081,
5501,
5581,
5801,
5851,
6469,
6949,
8501,
9001,
9049,
9221,
9551,
9649,
9851,
9949,
20021,
20201,
50207,
60649,
80051,
666649,
946669,
5200007,
22000001,
60000049,
66000049,
66600049,
80555551,
555555555551,
5000000000000000000000000000027}
11
{12,
16,
18,
21,
27,
29,
34,
38,
3A,
43,
49,
54,
56,
61,
65,
67,
72,
76,
81,
89,
92,
94,
98,
9A,
A3,
10A,
115,
117,
133,
139,
153,
155,
171,
193,
197,
199,
1AA,
225,
232,
236,
25A,
263,
315,
319,
331,
335,
351,
353,
362,
373,
379,
391,
395,
407,
414,
452,
458,
478,
47A,
485,
4A5,
4A7,
502,
508,
511,
513,
533,
535,
539,
551,
571,
579,
588,
595,
623,
632,
70A,
711,
715,
731,
733,
737,
755,
759,
775,
791,
797,
7AA,
803,
847,
858,
85A,
874,
885,
887,
913,
919,
931,
937,
957,
959,
975,
995,
A07,
A1A,
A25,
A45,
A74,
A7A,
A85,
AA1,
AA7,
1101,
11A9,
1305,
1451,
1457,
15A7,
175A,
17A5,
17A9,
2023,
2045,
2052,
2083,
20A5,
2333,
2A05,
2A52,
3013,
3026,
3059,
3097,
3206,
3222,
3233,
3307,
3332,
3505,
4025,
4151,
4157,
4175,
4405,
4445,
4487,
450A,
4575,
5017,
5031,
5059,
5075,
5097,
5099,
5105,
515A,
517A,
520A,
5301,
5583,
5705,
577A,
5853,
5873,
5909,
5A17,
5A57,
5A77,
5A8A,
6683,
66A9,
7019,
7073,
7079,
7088,
7093,
7095,
7309,
7451,
7501,
7507,
7578,
757A,
75A7,
7787,
7804,
7844,
7848,
7853,
7877,
78A4,
7A04,
7A57,
7A79,
7A95,
8078,
8245,
8333,
8355,
8366,
8375,
8425,
8553,
8663,
8708,
8777,
878A,
8A05,
9053,
9101,
9107,
9305,
9505,
9703,
A052,
A119,
A151,
A175,
A515,
A517,
A575,
A577,
A5A8,
A719,
A779,
A911,
AAA9,
10011,
10075,
10091,
10109,
10411,
10444,
10705,
10709,
10774,
10901,
11104,
11131,
11144,
11191,
1141A,
114A1,
13757,
1411A,
14477,
144A4,
14A04,
14A11,
17045,
17704,
1774A,
17777,
177A4,
17A47,
1A091,
1A109,
1A114,
1A404,
1A411,
1A709,
20005,
20555,
22203,
25228,
25282,
25552,
25822,
28522,
30037,
30701,
30707,
31113,
33777,
35009,
35757,
39997,
40045,
4041A,
40441,
4045A,
404A1,
4111A,
411A1,
42005,
44401,
44474,
444A1,
44555,
44577,
445AA,
44744,
44A01,
47471,
47477,
47701,
5057A,
50903,
5228A,
52A22,
52A55,
52A82,
55007,
550A9,
55205,
55522,
55557,
55593,
55805,
57007,
57573,
57773,
57807,
5822A,
58307,
58505,
58A22,
59773,
59917,
59973,
59977,
59999,
5A015,
5A2A2,
5AA99,
60836,
60863,
68636,
6A609,
6A669,
6A696,
6A906,
6A966,
70048,
70103,
70471,
70583,
70714,
71474,
717A4,
71A09,
74084,
74444,
74448,
74477,
744A8,
74747,
74774,
7488A,
74A48,
75773,
77144,
77401,
77447,
77799,
77A09,
78008,
78783,
7884A,
78888,
788A8,
79939,
79993,
79999,
7A051,
7A444,
7A471,
80005,
80252,
80405,
80522,
80757,
80AA5,
83002,
84045,
85307,
86883,
88863,
8A788,
90073,
90707,
90901,
95003,
97779,
97939,
99111,
99177,
99973,
A0111,
A0669,
A0966,
A0999,
A0A09,
A1404,
A4177,
A4401,
A4717,
A5228,
A52AA,
A5558,
A580A,
A5822,
A58AA,
A5A59,
A5AA2,
A6096,
A6966,
A6999,
A7051,
A7778,
A7808,
A9055,
A9091,
A9699,
A9969,
AA52A,
AA58A,
100019,
100079,
101113,
101119,
101911,
107003,
140004,
144011,
144404,
1A0019,
1A0141,
1A5001,
1A7005,
1A9001,
222223,
222823,
300107,
300202,
300323,
303203,
307577,
310007,
332003,
370777,
400555,
401A11,
404001,
404111,
405AAA,
41A011,
440A41,
441011,
451777,
455555,
470051,
470444,
474404,
4A0401,
4A4041,
500015,
500053,
500077,
500507,
505577,
522A2A,
525223,
528A2A,
531707,
550777,
553707,
5555A9,
555A99,
557707,
55A559,
5807A7,
580A0A,
580A55,
58A0AA,
590007,
599907,
5A2228,
5A2822,
5A2AAA,
5A552A,
5AA22A,
5AAA22,
60A069,
683006,
6A0096,
6A0A96,
6A9099,
6A9909,
700778,
701074,
701777,
704408,
704417,
704457,
704484,
707041,
707441,
707708,
707744,
707784,
710777,
717044,
717077,
740008,
74484A,
770441,
770744,
770748,
770771,
777017,
777071,
777448,
777484,
777701,
7778A8,
777A19,
777A48,
778883,
78A808,
790003,
7A1009,
7A4408,
7A7708,
80A555,
828283,
828883,
840555,
850505,
868306,
873005,
883202,
900701,
909739,
909979,
909991,
970771,
977701,
979909,
990739,
990777,
990793,
997099,
999709,
999901,
A00009,
A00599,
A01901,
A05509,
A0A058,
A0A955,
A10114,
A555A2,
A55999,
A59991,
A5A222,
A5A22A,
A60609,
A66069,
A66906,
A69006,
A79005,
A87888,
A90099,
A90996,
A96006,
A96666,
A97177,
A97771,
AA0A58,
AA5A22,
AAA522,
1000501,
1011141,
1030007,
1070047,
111114A,
1111A14,
1111A41,
1144441,
14A4444,
1700005,
1700474,
1A44444,
2555505,
2845055,
3030023,
3100003,
3333397,
4000111,
4011111,
41A1111,
4411111,
444441A,
4444771,
4470004,
4505005,
4744417,
4774441,
4777404,
4777417,
4777747,
4A11111,
4A40001,
5000093,
50005A7,
5005777,
5050553,
5055503,
5070777,
5222222,
5222AAA,
52AAAA2,
52AAAAA,
5505053,
5552AAA,
5555599,
5555A58,
5558A0A,
5558A55,
5558AAA,
55A0009,
55AAA52,
580000A,
5822222,
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D0(B^^17804),
D(B^^32234),
(4^^72785)DD,
(3^^116137)AF}

I have solved bases b = 2 up to 16 i.e. I have found all such primes
(elements in the minimal set of the primes > b in base b) and proved
that these are all such primes.

For 8 families only strong probable primes are known:

bfamily
(base b form)		family
(algebraic form)		strong probable
prime (base b form)		strong probable
prime (algebraic form)
115(7^^n)(57*11^n-7)/105(7^^62668)(57*11^62668-7)/10
138(0^^n)1118*13^(n+3)+1838(0^^32017)1118*13^32020+183
139(5^^n)(113*13^n-5)/129(5^^197420)(113*13^197420-5)/12
13A(3^^n)A(41*13^(n+1)+27)/4A(3^^592197)A(41*13^592198+27)/4
13C(5^^n)C(149*13^(n+1)+79)/12C(5^^23755)C(149*13^23756+79)/12
16(3^^n)AF(16^(n+2)+619)/5(3^^116137)AF(16^116139+619)/5
16(4^^n)DD(4*16^(n+2)+2291)/15(4^^72785)DD(4*16^72787+2291)/15
16D(B^^n)(206*16^n-11)/15D(B^^32234)(206*16^32234-11)/15

It can be noted that the now solved family A(3^^n)A in base 13
is "Plateau and Depression" family
( see http://www.worldofnumbers.com/deplat.htm (only base 10) ).
Also
two large primes in the base 13 set (when written in base 13)
are also Plateau and Depression Primes (PDP's):
7(5^^375)7 (proven prime) and C(5^^23755)C (probable prime).
These two primes are the first primes in families
7(5^^n)7 and C(5^^n)C in base 13, these two families
(as well as the now solved family A(3^^n)A in base 13) are
"Plateau and Depression" families that have no small primes,
like the status of the family 7(1^^n)7 in base 10, its first
prime is 7(1^^10905)7, see deplat.htm#pdp717.

“ To find the sets of various bases,
we have to determine whether family x(d^^n)y in base b
(where x and y are base b strings (may be empty),
and d is a base b digit) contains a prime
(only count the numbers > base) or not.”
a quote from Xinyao Chen.

Why some families only contain composites

Example 1: base 10, family 4(6^^n)9 (n⩾0)

All numbers are divisible by 7

Example 2: base 9, family 5(1^^n)
(n⩾1, since we only consider the numbers > base)
(the same as WONplate 219, base 9 appending 1's with k = 5)

n == 1 mod 2: factor of 2
n == 0 mod 2: factor of 5

Example 3: base 11, family 2(5^^n)
(n⩾1, since we only consider the numbers > base)

n == 0 mod 2: factor of 2
n == 1 mod 2: factor of 3

Example 4: base 8, family 6(4^^n)7 (n⩾0)

n == 1 mod 2: factor of 3
n == 0 mod 4: factor of 5
n == 2 mod 4: factor of 13

Example 5: base 13, family 3(0^^n)95 (n⩾0)

n == 1 mod 2: factor of 7
n == 2 mod 4: factor of 5
n == 0 mod 4: factor of 17

Example 6: base 16, family (4^^n)D
(n⩾1, since we only consider the numbers > base)

n == 0 mod 3: factor of 13
n == 1 mod 3: factor of 7
n == 2 mod 3: factor of 3

Example 7: base 9, family (1^^n)
(n⩾2, since we only consider the numbers > base)
(the same as WONplate 219, base 9 appending 1's with k = 1)

The formula is (9^n-1)/8,
which can be factored as (3^n-1) * (3^n+1) / 8,
and since if n⩾3, 3^n-1 ⩾ 3^3-1 = 26 > 8, 3^n+1 ⩾ 3^3+1 = 28 > 8,
this factorization is nontrivial if n⩾3,
and this only remains to check the case n=2, but for n=2,
(9^n-1)/8 = 10 and 10 is not prime, thus no number of this form is prime.

Example 8: base 9, family 3(8^^n)
(n⩾1, since we only consider the numbers > base)
(the same as WONplate 219, base 9 appending 8's with k = 3)

The formula is 4*9^n-1,
which can be factored as (2*3^n-1) * (2*3^n+1),
and since if n⩾1,
2*3^n-1 ⩾ 2*3^1-1 = 5 > 1, 2*3^n+1 ⩾ 2*3^1+1 = 7 > 1,
this factorization is nontrivial, thus no number of this form is prime.

Example 9: base 8, family 1(0^^n)1 (n⩾0)

The formula is 8^(n+1)+1,
which can be factored as (2^(n+1)+1) * (4^(n+1)-2^(n+1)+1),
and since if n⩾0,
2^(n+1)+1 ⩾ 2^1+1 = 3 > 1, 4^(n+1)-2^(n+1)+1 ⩾ 4^1-2^1+1 = 3 > 1,
this factorization is nontrivial, thus no number of this form is prime.

Example 10: base 16, family (4^^n)1
(n⩾1, since we only consider the numbers > base)

The formula is (4*16^(n+1)-49)/15,
which can be factored as (2*4^(n+1)-7) * (2*4^(n+1)+7) / 15,
and since if n⩾1,
2*4^(n+1)-7 ⩾ 2*4^2-7 = 25 > 15, 2*4^(n+1)+7 ⩾ 2*4^2+7 = 39 > 15,
this factorization is nontrivial, thus no number of this form is prime.

Example 11: base 12, family (B^^n)9B (n⩾0)

The formula is 12^(n+2)-25

n == 1 mod 2: factor of 13
n == 0 mod 2: let n = 2*m, 12^(n+2)-25 = 12^(2*m+2)-25 =
(12^(m+1)-5) * (12^(m+1)+5),
and since if n⩾0, then m is also ⩾ 0,
and 12^(m+1)-5 ⩾ 12^1-5 = 7 > 1, 12^(m+1)+5 ⩾ 12^1+5 = 17 > 1,
this factorization is nontrivial, thus no number of this form is prime.

Example 12: base 14, family 8(D^^n)
(n⩾1, since we only consider the numbers > base)

The formula is 9*14^n-1

n == 1 mod 2: factor of 5
n == 0 mod 2: let n = 2*m, 9*14^n-1 = 9*14^(2*m)-1 =
(3*14^m-1) * (3*14^m+1), and since if n⩾1,
then m is also ⩾ 1, and 3*14^m-1 ⩾ 3*14^1-1 = 41 > 1,
3*14^m+1 ⩾ 3*14^1+1 = 43 > 1,
this factorization is nontrivial, thus no number of this form is prime.
“ Some families x(d^^n)y could not be ruled out as containing no primes > b,
but no primes > b could be found in the family, even after searching through
numbers with over 50000 digits.
Many families x(d^^n)y contain no small primes (only count the numbers > base)
even though they do contain very large primes.”
by Xinyao Chen.

Examples for families that cannot be ruled out as only
contain composites but contain no small primes
(only count the numbers > base)

Example 1: base 10, family 5(0^^n)27 (n⩾0)

The smallest prime is 5(0^^28)27

Example 2: base 5, family 1(0^^n)13 (n⩾0)

The smallest prime is 1(0^^93)13

Example 3: base 8, family (4^^n)7
(n⩾1, since we only consider the numbers > base)

The smallest prime is (4^^220)7

Example 4: base 9, family 3(0^^n)11 (n⩾0)

The smallest prime is 3(0^^1158)11

Example 5: base 14, family 4(D^^n)
(n⩾1, since we only consider the numbers > base)

The smallest prime is 4(D^^19698)
(this prime can be easily proven prime using N+1 primality test,
since N+1 is trivially factored)

Example 6: base 13, family 8(0^^n)111 (n⩾0)

The smallest (probable) prime is 8(0^^32017)111
(this PRP has 35670 decimal digits, and because of its size
and neither N-1 nor N+1 is trivially factored,
this number is not easily proven prime)

Example 7: base 16, family D(B^^n)
(n⩾1, since we only consider the numbers > base)

The smallest (probable) prime is D(B^^32234)
(this PRP has 38815 decimal digits, and because of its size
and neither N-1 nor N+1 is trivially factored,
this number is not easily proven prime)

Example 8: base 11, family 5(7^^n)
(n⩾1, since we only consider the numbers > base)

The smallest (probable) prime is 5(7^^62668)
(this PRP has 65263 decimal digits, and because of its size
and neither N-1 nor N+1 is trivially factored,
this number is not easily proven prime) 

Example 9: base 16, family (4^^n)DD (n⩾0)

The smallest (probable) prime is (4^^72785)DD
(this PRP has 87644 decimal digits, and because of its size
and neither N-1 nor N+1 is trivially factored,
this number is not easily proven prime)

Example 10: base 16, family (3^^n)AF (n⩾0)

The smallest (probable) prime is (3^^116137)AF
(this PRP has 139845 decimal digits, and because of its size
and neither N-1 nor N+1 is trivially factored,
this number is not easily proven prime)



Condensed table: ( '^^' is symbol for concatenation )




base bnumber of
elements
in the
set of
base b		the largest
element in
the set of
base b in
'base b' form		the largest
element in
the set of
base b in
'algebraic' form		length of
the largest
element
in the set
of base b


211132


33111133


45221413


5221(0^^93)135^95+896


6114004152095


771(3^^16)1(7^17-5)/217


875(4^^220)7(4*8^221+17)/7221


91513(0^^1158)113*9^1160+101161


10775(0^^28)275*10^30+2731


1110685(7^^62668)(57*11^62668-7)/1062669


121064(0^^39)774*12^41+9142


133197A(3^^592197)A(41*13^592198+27)/4592199


146504(D^^19698)5*14^19698-119699


151284(7^^155)97(15^157+59)/2157


162347(3^^116137)AF(16^116139+619)/5116139




Note:
The data for base 11 assumes that the PRP 5(7^^62668) is actual prime.
The data for base 13 assumes that the PRP's C(5^^23755)C,
   8(0^^32017)111, 9(5^^197420) and A(3^^592197)A are actual primes.
The data for base 16 assumes that the PRP's D(B^^32234),
   (4^^72785)DD and (3^^116137)AF are actual primes.



Primality certificates for primes > 10^299 in these sets:

Base 9:

7(6^^329)2: http://factordb.com/cert.php?id=1100000002359003642
2(7^^686)07: http://factordb.com/cert.php?id=1100000002495467486
3(0^^1158)11: http://factordb.com/cert.php?id=1100000002376318423

(These enabled us to complete the classification of the minimal elements
of primes > b in base b, for all 2 ⩽ b ⩽ 10)

Base 11:

(A^^713)58: http://factordb.com/cert.php?id=1100000003576826487
(7^^759)44: http://factordb.com/cert.php?id=1100000002505568840
55(7^^1011): http://factordb.com/cert.php?id=1100000002361376522

( 5(7^^62668) is only PRP, it is strong PRP to bases 2, 3, 5, 7, 11, 13,
  17, 19, 23 and trial factored to 10^11, verified by PFGW )

Base 13:

5(0^^270)44: http://factordb.com/cert.php?id=1100000002632397005
(9^^271)095: http://factordb.com/cert.php?id=1100000003590431654
1(0^^286)7771: http://factordb.com/cert.php?id=1100000003590431633
(9^^308)1: (proven prime by N-1 primality test,
factorization of N-1 (equivalent to factor the number 13^308-1)
(B^^341)C4: http://factordb.com/cert.php?id=1100000003590431618
8(B^^343): http://factordb.com/cert.php?id=1100000002321018736
71(0^^371)111: http://factordb.com/cert.php?id=1100000003590431609
7(5^^375)7: http://factordb.com/cert.php?id=1100000003590431596
9B(0^^391)9: http://factordb.com/cert.php?id=1100000002632396790
7B0(B^^397): http://factordb.com/cert.php?id=1100000003590431574
1(0^^414)93: http://factordb.com/cert.php?id=1100000002523249240
8101(0^^415)1: (can be easily proven prime with the N-1 primality test,
since its N-1 is trivially 100% factored)
811(0^^435)1: (can be easily proven prime with the N-1 primality test,
since its N-1 is trivially 100% factored)
B(7^^486): http://factordb.com/cert.php?id=1100000002321015892
(B^^563)C: (proven prime by N-1 primality test,
factorization of N-1 (equivalent to factor the number 13^564-1)
1(B^^576): (proven prime by N-1 primality test,
factorization of N-1 (equivalent to factor the number 13^576-1)
8(0^^693)87: (proven prime by N-1 primality test,
factorization of N-1 and
primality certificate of large prime factor of N-1)
CC(5^^713): http://factordb.com/cert.php?id=1100000002615627353
(B^^834)74: http://factordb.com/cert.php?id=1100000003590430871
(9^^968)B: http://factordb.com/cert.php?id=1100000000258566244
1(0^^1295)181: http://factordb.com/cert.php?id=1100000002615445013
(9^^1362)5: http://factordb.com/cert.php?id=1100000002321017776
(7^^1504)1: http://factordb.com/cert.php?id=1100000002320890755
93(0^^1551)1: (can be easily proven prime with the N-1 primality test,
since its N-1 is trivially 100% factored)
72(0^^2297)2: http://factordb.com/cert.php?id=1100000002632396910
177(0^^2703)17: http://factordb.com/cert.php?id=1100000003590430825
39(0^^6266)1: (can be easily proven prime with the N-1 primality test,
since its N-1 is trivially 100% factored)
B(0^^6540)BBA: http://factordb.com/cert.php?id=1100000002616382906
(C^^10631)92: http://factordb.com/cert.php?id=1100000003590493750

( C(5^^23755)C, 8(0^^32017)111, 9(5^^197420) and A(3^^592197)A are only
  PRPs, they are strong PRP to bases 2, 3, 5, 7, 11, 13, 17, 19, 23
  and trial factored to 10^11, verified by PFGW )

Base 14:

The base 14 primes 4(D^^19698) and 34(D^^708) can be easily proven prime
with the N+1 primality test, since their N+1 are trivially 100% factored.

Base 16:

88(0^^246)7: (proven prime by N+1 primality test, factorization of N+1
(N+1 has no prime factors > 10^299)
D(4^^263)D: http://factordb.com/cert.php?id=1100000002468170238
E(0^^261)4DD: http://factordb.com/cert.php?id=1100000003588388352
8C(0^^290)ED: http://factordb.com/cert.php?id=1100000003588388307
D(A^^305)5: http://factordb.com/cert.php?id=1100000003588388284
CE8(0^^422)D: http://factordb.com/cert.php?id=1100000003588388257
5(F^^544)6F: http://factordb.com/cert.php?id=1100000002604723967
88(F^^545): (can be easily proven prime with the N+1 primality test,
since its N+1 is trivially 100% factored)
BE(0^^792)BB: http://factordb.com/cert.php?id=1100000003588387938
D(9^^1052): http://factordb.com/cert.php?id=1100000002321036020
FA(F^^1062)45: http://factordb.com/cert.php?id=1100000003588387610
F(8^^1517)F: http://factordb.com/cert.php?id=1100000000633744824
2(0^^1713)321: http://factordb.com/cert.php?id=1100000003588386735
300(F^^1960)AF: http://factordb.com/cert.php?id=1100000003588368750
9(0^^3542)91: http://factordb.com/cert.php?id=1100000000633424191
5B(C^^3700)D: http://factordb.com/cert.php?id=1100000000993764322
D0(B^^17804): http://factordb.com/cert.php?id=1100000003589278511

( D(B^^32234), (4^^72785)DD and (3^^116137)AF are only PRPs,
  they are strong PRP to bases 2, 3, 5, 7, 11, 13, 17, 19, 23
  and trial factored to 10^11, verified by PFGW )



Proof for Base 10

Chen has a proof for base 10, using the notation in Shallit's article.
He left it as an exercise for the reader to write the proof for bases
2 ⩽ b ⩽ 9 and base b=12. The proof for base b=9 is more complex,
e.g. we need to show that (3^^i)(0^^j)5 cannot be prime
(only count the numbers > base, i.e. i cannot be 0), since it is
divisible by 2 if i is odd and divisible by 5 if i is even.
For other bases b such as b=11 it is almost impossible to write the
proof, since for base b=11 there are too many (1068) minimal primes.


Theorem: If S = {primes > 10 (i.e. primes with length 2 or more)}, then
M(S) = {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521,
557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087,
2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049,
9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649,
946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551,
555555555551, 5000000000000000000000000000027}.


Proof: Assume p is a prime > 10 (i.e. p is a prime with length 2 or
more), and the last digit of p must lie in {1,3,7,9}.

Case 1: p ends with 1.

In this case we can write p = x1. If x contains 1, 3, 4, 6, or 7, then
(respectively) 11 ◁ p, 31 ◁ p, 41 ◁ p, 61 ◁ p, or 71 ◁ p.
Hence we may assume that x ∈ {2,5,8,9}{0,2,5,8,9}*.

Case 1.1: p begins with 2.

In this case we can write p = 2y1. If 5 ◁ y, then 251 ◁ p.
If 8 ◁ y, then 281 ◁ p. If 9 ◁ y, then 29 ◁ p.
Hence we may assume y ∈ {0,2}*.

If 22 ◁ y, then 2221 ◁ p.
Hence we may assume y contains zero or one 2's.

If y contains no 2's, then p ∈ 20*1. But then, since the sum of
the digits of p is 3, p is divisible by 3, so p cannot be prime.

If y contains exactly one 2, then we can write p = 2z2w1, where z,w ∈ 0*.
If 0 ◁ z and 0 ◁ w, then 20201 ◁ p.
Hence we may assume either z or w is empty.

If z is empty, then p ∈ 220*1,
and the smallest prime p ∈ 220*1 is 22000001.

If w is empty, then p ∈ 20*21,
and the smallest prime p ∈ 20*21 is 20021.

Case 1.2: p begins with 5.

In this case we can write p = 5y1. If 2 ◁ y, then 521 ◁ p.
If 9 ◁ y, then 59 ◁ p. Hence we may assume y ∈ {0,5,8}*.

If 05 ◁ y, then 5051 ◁ p. If 08 ◁ y, then 5081 ◁ p.
If 50 ◁ y, then 5501 ◁ p. If 58 ◁ y, then 5581 ◁ p.
If 80 ◁ y, then 5801 ◁ p. If 85 ◁ y, then 5851 ◁ p.
Hence we may assume y ∈ 0* ∪ 5* ∪ 8*.

If y ∈ 0*, then p ∈ 50*1. But then, since the sum of the digits
of p is 6, p is divisible by 3, so p cannot be prime.

If y ∈ 5*, then p ∈ 55*1, and the smallest
prime p ∈ 55*1 is 555555555551.

If y ∈ 8*, since if 88 ◁ y, then 881 ◁ p, hence we may
assume y ∈ {ε,8}, and thus p ∈ {51,581},
but 51 and 581 are both composite.

Case 1.3: p begins with 8.

In this case we can write p = 8y1. If 2 ◁ y, then 821 ◁ p.
If 8 ◁ y, then 881 ◁ p. If 9 ◁ y, then 89 ◁ p.
Hence we may assume y ∈ {0,5}*.

If 50 ◁ y, then 8501 ◁ p. Hence we may assume y ∈ 0*5*.

If 005 ◁ y, then 80051 ◁ p. Hence we may assume y ∈ 0* ∪ 5* ∪ 05*.

If y ∈ 0*, then p ∈ 80*1. But then, since the sum of the digits
of p is 9, p is divisible by 3, so p cannot be prime.

If y ∈ 5*, since if 55555555555 ◁ y, then 555555555551 ◁ p,
hence we may assume y ∈ {ε,5,55,555,5555,55555,555555,5555555,55555555,
555555555,5555555555}, and thus p ∈ {81,851,8551,85551,855551,8555551,
85555551,855555551,8555555551,85555555551,855555555551},
but all of these numbers are composite.

If y ∈ 05*, since if 55555555555 ◁ y, then 555555555551 ◁ p,
hence we may assume y ∈ {0,05,055,0555,05555,055555,0555555,05555555,055555555,
0555555555,05555555555}, and thus p ∈ {801,8051,80551,805551,8055551,80555551,
805555551,8055555551,80555555551,805555555551,8055555555551}, and of these
numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551,
thus only 80555551 is minimal prime.

Case 1.4: p begins with 9.

In this case we can write p = 9y1. If 9 ◁ y, then 991 ◁ p.
Hence we may assume y ∈ {0,2,5,8}*.

If 00 ◁ y, then 9001 ◁ p. If 22 ◁ y, then 9221 ◁ p.
If 55 ◁ y, then 9551 ◁ p. If 88 ◁ y, then 881 ◁ p.
Hence we may assume y contains at most one 0, at most one 2, at most one 5,
and at most one 8.

If y only contains at most one 0 and does not contain any of {2,5,8},
then y ∈ {ε,0}, and thus p ∈ {91,901}, but 91 and 901 are both composite.
If y only contains at most one 0 and only one of {2,5,8}, then the sum of
the digits of p is divisible by 3, p is divisible by 3, so p cannot be prime.
Hence we may assume y contains at least two of {2,5,8}.

If 25 ◁ y, then 251 ◁ p. If 28 ◁ y, then 281 ◁ p.
If 52 ◁ y, then 521 ◁ p. If 82 ◁ y, then 821 ◁ p.
Hence we may assume y contains no 2's (since if y contains 2,
then y cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ y, then 9851 ◁ p. Hence we may assume y ∈ {58,580,508,058},
and thus p ∈ {9581,95801,95081,90581}, and of these numbers only 95801
is prime, but 95801 is not minimal prime since 5801 ◁ 95801.

Case 2: p ends with 3.

In this case we can write p = x3. If x contains 1, 2, 4, 5, 7, or 8, then
(respectively) 13 ◁ p, 23 ◁ p, 43 ◁ p, 53 ◁ p, 73 ◁ p, or 83 ◁ p.
Hence we may assume that x ∈ {3,6,9}{0,3,6,9}*,
and thus p ∈ {3,6,9}{0,3,6,9}*3. But then, since the digits of p
all have a common factor 3, p is divisible by 3, so p cannot be prime.

Case 3: p ends with 7.

In this case we can write p = x7. If x contains 1, 3, 4, 6, or 9, then
(respectively) 17 ◁ p, 37 ◁ p, 47 ◁ p, 67 ◁ p, or 97 ◁ p.
Hence we may assume that x ∈ {2,5,7,8}{0,2,5,7,8}*.

Case 3.1: p begins with 2.

In this case we can write p = 2y7. If 2 ◁ y, then 227 ◁ p.
If 5 ◁ y, then 257 ◁ p. If 7 ◁ y, then 277 ◁ p.
Hence we may assume y ∈ {0,8}*.

If 08 ◁ y, then 2087 ◁ p. If 88 ◁ y, then 887 ◁ p.
Hence we may assume y ∈ 0* ∪ 80*.

If y ∈ 0*, then p ∈ 20*7. But then, since the sum of the digits
of p is 9, p is divisible by 3, so p cannot be prime.

If y ∈ 80*, then p ∈ 280*7. But then p is divisible by 7,
since for i ⩾ 0 we have 7 ⋅ 4(0^^i)1 = 28(0^^i)7.

Case 3.2: p begins with 5.

In this case we can write p = 5y7. If 5 ◁ y, then 557 ◁ p.
If 7 ◁ y, then 577 ◁ p. If 8 ◁ y, then 587 ◁ p.
Hence we may assume y ∈ {0,2}*.

If 22 ◁ y, then 227 ◁ p.
Hence we may assume y contains zero or one 2's.

If y contains no 2's, then p ∈ 50*7. But then, since the sum of the digits
of p is 12, p is divisible by 3, so p cannot be prime.

If y contains exactly one 2, then we can write p = 5z2w7, where z,w ∈ 0*.
If 0 ◁ z and 0 ◁ w, then 50207 ◁ p.
Hence we may assume either z or w is empty.

If z is empty, then p ∈ 520*7, and the smallest prime p ∈ 520*7 is 5200007.

If w is empty, then p ∈ 50*27, and the smallest prime p ∈ 50*27
is 5000000000000000000000000000027.

Case 3.3: p begins with 7.

In this case we can write p = 7y7. If 2 ◁ y, then 727 ◁ p.
If 5 ◁ y, then 757 ◁ p. If 8 ◁ y, then 787 ◁ p.
Hence we may assume that y ∈ {0,7}*, and thus p ∈ 7{0,7}*7.
But then, since the digits of p all have a common factor 7, p is divisible
by 7, so p cannot be prime.

Case 3.4: p begins with 8.

In this case we can write p = 8y7. If 2 ◁ y, then 827 ◁ p.
If 5 ◁ y, then 857 ◁ p. If 7 ◁ y, then 877 ◁ p.
If 8 ◁ y, then 887 ◁ p. Hence we may assume y ∈ 0*, and thus p ∈ 80*7.
But then, since the sum of the digits of p is 15, p is divisible by 3,
so p cannot be prime.

Case 4: p ends with 9.

In this case we can write p = x9. If x contains 1, 2, 5, 7, or 8, then
(respectively) 19 ◁ p, 29 ◁ p, 59 ◁ p, 79 ◁ p, or 89 ◁ p.
Hence we may assume x ∈ {3,4,6,9}{0,3,4,6,9}*.

If 44 ◁ x, then 449 ◁ p. Hence we may assume x contains zero or one 4's.

If x contains no 4's, then x ∈ {3,6,9}{0,3,6,9}*,
and thus p ∈ {3,6,9}{0,3,6,9}*9. But then, since the digits of p
all have a common factor 3, p is divisible by 3, so p cannot be prime.
Hence we may assume that x contains exactly one 4.

Case 4.1: p begins with 3.

In this case we can write p = 3y4z9, and we must have 349 ◁ p.

Case 4.2: p begins with 4.

In this case we can write p = 4y9, where y ∈ {0,3,6,9}*. If 0 ◁ y,
then 409 ◁ p. If 3 ◁ y, then 43 ◁ p. If 9 ◁ y, then 499 ◁ p.
Hence we may assume y ∈ 6*, and thus p ∈ 46*9. But then p is divisible by 7,
since for i ⩾ 0 we have 7 ⋅ (6^^i)7 = 4(6^^i)9.

Case 4.3: p begins with 6.

In this case we can write p = 6y4z9, where y,z ∈ {0,3,6,9}*. If 0 ◁ z,
then 409 ◁ p. If 3 ◁ z, then 43 ◁ p. If 6 ◁ z, then 6469 ◁ p.
If 9 ◁ z, then 499 ◁ p. Hence we may assume z is empty.

If 3 ◁ y, then 349 ◁ p. If 9 ◁ y, then 6949 ◁ p.
Hence we may assume y ∈ {0,6}*.

If 06 ◁ y, then 60649 ◁ p. Hence we may assume y ∈ 6*0*.

If 666 ◁ y, then 666649 ◁ p. If 00000 ◁ y, then 60000049 ◁ p.
Hence we may assume y ∈ {ε,6,66}{ε,0,00,000,0000},
and thus p ∈ 6{ε,6,66}{ε,0,00,000,0000}49,
and of these numbers only 66000049 and 66600049 are primes.

Case 4.4: p begins with 9.

In this case we can write p = 9y4z9, where y,z ∈ {0,3,6,9}*.
If 0 ◁ y, then 9049 ◁ p. If 3 ◁ y, then 349 ◁ p.
If 6 ◁ y, then 9649 ◁ p. If 9 ◁ y, then 9949 ◁ p.
Hence we may assume y is empty.

If 0 ◁ z, then 409 ◁ p. If 3 ◁ z, then 43 ◁ p.
If 9 ◁ z, then 499 ◁ p. Hence we may assume z ∈ 6*,
and thus p ∈ 946*9, and the smallest prime p ∈ 946*9 is 946669.


A000218 Prime Curios! Prime Puzzle
Wikipedia 218 Le nombre 218

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