## Composite Sequences

In the February 2011 Mathematical Monthly Lenny Jones [1] considered the following question:
For a given digit d determine the smallest value k relatively prime to d so that when the digit d is repeatedly appended to the end of k the sequence kd, kdd, kddd, ... contains only composite numbers.
Jones proved that for d=1 the smallest k is 37. and noted that for d=3, k <= 4070, d=7, k <= 606474, and d=9, k <= 1879711.
Grantham, Jarnicki, Rickert and Wagon (GJRW, to appear) determined that for d=3, k = 4070 or 817, d=7, k = 891, d=9, k = 10175 or 8578 or 7018 or 4420 and conjectured that k3 = 4070, k9 = 10175.
In the same paper GJRW studied the question in other bases and determined answers for several cases and conjectures for most of the bases less than ten.
For each base b and trailing digit d relatively prime to the base this table lists the smallest known seed k so that appending the digit d any positive integer number of times to the base b representation of k always produces a composite number. The first column is the base b, the second column is the digit appended d, the third column is the base ten representation of the smallest known seed k, the fourth column is the base b representation of the smallest known seed, the fifth column is the number of values relatively prime to d that are less than k for which no prime is yet known for the sequence.
 Base Digit seed base 10 seed base b Candidates 2 1 509202 1111100010100010010(1) 58 H. Riesel 3 1 6059 22022102(1) 15 2 63064644937 20000210002020220021021(2) CRUS 4 1 5 11(1) Minimum 3 8 20(3) Minimum 5 1 3 3(1) Minimum 2 191115 22103430(2) 258 3 585655 122220110(3) 2274 4 346801 42044201(4) 126 6 1 26789 324005(1) 29 5 84686 1452022(5) 2 7 1 76 136(1) Minimum* 2 15979 64405(2) 17 3 5629 22261(3) 19 4 20277 113055(4) 16 5 43 2211(5) Minimum 6 408034255081 41323316641135(6) CRUS 8 1 21 25(1) Minimum 3 1079770 4074732(3) 721 5 7476 16464(5) 29 7 13 15(7) Minimum 9 1 1 1(1) Minimum 2 4615 6287(2) 14 4 6059 8272(4) 12 5 78 86(5) Minimum* 7 2 2(7) Minimum 8 3 3(8) Minimum 10 1 37 37(1) Minimum L. Jones 3 4070 4070(3) 1 L. Jones, S. Wagon 7 891 891(7) Minimum* S. Wagon 9 10175 10175(9) 3 CRUS
* In these cases some of the primes found for smaller seeds are probable primes that have not been certified.
H. Riesel, Några stora primtal (Swedish: Some large primes), Elementa 39 (1956), 258-260.
L. Jones, When does appending the same digit repeatedly on the right of a positive integer generate a sequence of composite numbers?, Amer. Math. Monthly 118 (2011), 153-160.

#### Base 3

##### Digit 1

The prime factors of 6059 follow a pattern of period 12 using the cover
 Residue 1 2 2 4 0 Modulus 2 3 4 6 12 Prime 2 13 5 7 73
The primes dividing 12A+r are
 0 1 2 3 4 5 6 7 8 9 10 11 73 2 5 2 7 2 5 2 13 2 5 2
In list form {73,2,5,2,7,2,5,2,13,2,5,2}.
Candidates checked to 19200 digits.
(806,2419), (915,2746), 2877, (968,2905), 3059, 3393, 3689, 3738, 3755, 3813, 3969, 4029, (156, 469, 1408, 4225), 4356, 4388, (498, 1495, 4486), 4589, (1565,4696), 4905, 5325, (1794,5383), 5625, 5798, 5876, 6059.
Values with first prime occurring with more than 5000 digits.
k exponent k exponent k exponent
2055 12978, 2718 9567, 3059 28580
3158 15331 3368 17455 3515 5898
3755 26022 3788 5031 4225 24758
4376 16533 4486 20845 4589 21404
4655 11134 4819 8968 5005 12082
5625 24314 5759 11140
The remaining 17 values, and how far they've been checked. There are 15 candidates remaining through 50000 digits.
 value prime at composite to value prime at composite to (806,2419) 50000 4029 47256 (915,2746) 50000 4356 50000 (968,2905) 50000 4388 50000 (1565,4696) 50000 4905 50000 2877 50000 5325 50000 3393 50000 5383 50000 3738 50000 5798 50000 3813 50000 5876 36665 3969 50000

##### Digit 2
56532669941364262542767 has period 120 using the cover
 0 3 4 2 1 2 5 13 17 13 16 10 3 4 5 6 8 10 12 15 20 24 30 30 13 5 11 7 41 61 73 4561 1181 6481 31 271
The primes dividing 120A + r are
 r 0 1 2 3 4 5 6 7 8 9 prime 13 41 7 13 11 73 13 5 7 13 r 10 11 12 13 14 15 16 17 18 19 prime 271 5 13 4561 11 13 31 41 13 5 r 20 21 22 23 24 25 26 27 28 29 prime 7 13 61 5 13 41 7 13 4561 11 r 30 31 32 33 34 35 36 37 38 39 prime 13 5 7 13 11 5 13 1181 7 13 r 40 41 42 43 44 45 46 47 48 49 prime 271 41 13 5 11 13 31 5 13 11 r 50 51 52 53 54 55 56 57 58 59 prime 7 13 61 73 13 5 7 13 4561 5 r 60 61 62 63 64 65 66 67 68 69 prime 13 6481 7 13 11 41 13 5 7 13 r 70 71 72 73 74 75 76 77 78 79 prime 271 5 13 41 11 13 31 73 13 5 r 80 81 82 83 84 85 86 87 88 89 prime 7 13 61 5 13 6481 7 13 4561 11 r 90 91 92 93 94 95 96 97 98 99 prime 13 5 7 13 11 5 13 41 7 13 r 100 101 102 103 104 105 106 107 108 109 prime 271 73 13 5 11 13 31 5 13 11 r 110 111 112 113 114 115 116 117 118 119 prime 7 13 61 41 11 5 7 13 4561 5
As a list this is
{13,41,7,13,11,73,13,5,7,13,271,5,13,4561,11,13,31,41,13,5,7,13,61,5,13,41,7,13,4561,11, 13,5,7,13,11,5,13,1181,7,13,271,41,13,5,11,13,31,5,13,117,13,61,73,13,5,7,13,4561,5, 13,6481,7,13,11,41,13,5,7,13,271,5,13,41,11,13,31,73,13,5,7,13,61,5,13,6481,7,13,4561,11, 13,5,7,13,11,5,13,41,7,13,271,73,13,5,11,13,31,5,13,11,7,13,61,41,11,5,7,13,4561,5}
Up to 4*105 only eight values require at least 3000 digits
Value First Prime Composite to at leastFilter
954155933
1282714259
20188590003m+2,12m+6
2862475932
30109530003m,12m+10
34337130003m+1,24m+15
36778530003m+1,12m+6
38481530003m,36m+10

#### Base 4

##### Digit 1
sn4,1(5) = ((3*5+1)*4k-1)/3 = (4k+2-1)/3= (2k+2-1)(2k+2-1)/3.
 Seed First Prime Seed First Prime Seed First Prime Seed First Prime 1 1 2 2 3 1 4 1
Note that since sn4,1(1) = (2k+1-1)(2k+1-1)/3,   sn4,1(1) is prime only for n=1.
##### Digit 3
sn4,3(3) = ((3+1)*4k-1 = (4k+1-1)= (2k+1-1)(2k+1-1).
 Seed First Prime Seed First Prime 1 1 2 1

#### Base 5

##### Digit 1
The prime factors of sn5,1(3) have period 2
 Residue 1 0 Modulus 2 2 Prime 2 3
The prime divisor list is {3,2}.
 Seed First Prime Seed First Prime 1 2 2 1
##### Digit 2
sn5,2(191115) is covered using a period 12 cover
 Residue 0 0 3 1 5 Modulus 2 3 4 6 12 Prime 3 31 13 7 601
The primes dividing 12A+r are
 0 1 2 3 4 5 6 7 8 9 10 11 3 7 3 31 3 601 3 13 3 31 3 13
The list form of the prime divisors is {3,7,3,31,3,601,3,13,3,31,3,13}.
There are 258 candidates listed at Base5Digit2.html checked to 15000 digits.
##### Digit 3
sn5,3(585655) is covered using a period 12 cover
 Residue 1 2 2 0 4 Modulus 2 3 4 6 12 Prime 2 31 13 7 601
The primes dividing 12A+r are
 0 1 2 3 4 5 6 7 8 9 10 11 7 2 31 2 601 2 7 2 31 2 13 2
The list of primes is {7,2,31,2,601,2,7,2,31,2,13,2}.
There are 2274 candidates listed at Base5Digit3.html checked to 15000 digits.
##### Digit 4
sn5,4(346801) is covered using a period 12 cover
 Residue 1 2 0 0 4 Modulus 2 3 4 6 12 Prime 3 31 13 7 601
The primes dividing 12A+r are
 0 1 2 3 4 5 6 7 8 9 10 11 7 3 31 3 13 3 7 3 31 3 601 3
The list of primes is {7,3,31,3,13,3,7,3,31,3,601,3}.
There are 127 candidates listed at Base5Digit4.html checked to 15000 digits.

#### Base 6

##### Digit 1
sn6,1(26789) is covered using a period 12 cover
 Residue 0 0 3 1 5 Modulus 2 3 4 6 12 Prime 7 43 37 31 13
The primes dividing 12A+r are
 0 1 2 3 4 5 6 7 8 9 10 11 7 37 7 43 7 13 7 37 7 43 7 37
The list of primes is {7,31,7,43,7,13,7,37,7,43,7,37}.
There are 29 candidate seed values less than 26789 checked to 15000 digits.
 value prime at composite to value prime at composite to value prime at composite to 50 3008 8443 2700 18592 13000 98 3041 8877 3246 19503 2094 525 13000 9219 4015 19979 3780 575 6476 9288 3826 20518 3150 848 7056 9954 13000 20616 13000 1247 13000 10137 13000 20770 13000 1666 10461 10788 13000 21147 10851 1757 4091 10907 13000 21672 4279 1898 13000 11012 13000 21868 13000 2212 3225 11165 5103 22058 5566 3220 4009 11585 13000 22163 5130 4606 11921 12285 5009 22169 2467 5033 2769 13637 2694 22232 13000 5292 11657 14043 3206 22940 13000 5580 2578 14141 2548 23016 13000 5769 2504 15352 13000 23415 2109 5853 3020 15359 10630 23758 13000 6032 2890 15708 2939 23786 7031 6321 2147 15960 13000 24683 13000 6490 3826 16682 13000 25277 13000 6804 5807 16982 9k-13k 25494 13000 6966 6820 17243 13000 25788 13000 7406 13000 17675 5337 26230 3002 7575 8976 17759 13000 26363 2190 8127 13000 17997 4745 26495 13000
##### Digit 5
sn6,5(84686) is covered using a period 12 cover
 Residue 0 1 1 3 11 Modulus 2 4 6 12 12 Prime 7 37 31 13 97
The primes dividing 12A+r are
 0 1 2 3 4 5 6 7 8 9 10 11 7 37 7 13 7 37 7 31 7 37 7 97
The list of primes is {7,37,7,13,7,37,7,31,7,37,7,97}.
There are 95 values below 84686 checked to 2000 digits.
1596, 1834, 3589, 4396, 6872, 6986, 8258, 8743, 9366, 9576, 9581, 10694, 11009, 11142, 11389, 14502, 15064, 16828, 17106, 17458, 18284, 19388, 19467, 19906, 20683, 20713, 21539, 21798, 23791, 25051, 26381, 26899, 27447, 28098, 28539, 29846, 30086, 32667, 33626, 35007, 35964, 36771, 37294, 40656, 41237, 41921, 43629, 43993, 44324, 47388, 48384, 48949, 49462, 49553, 49791, 51016, 52463, 53093, 54957, 56201, 57022, 57461, 57491, 57561, 58576, 58756, 59094, 59568, 60648, 61894, 62377, 63028, 63798, 64169, 65002, 66038, 66059, 66302, 66857, 68339, 68857, 70166, 73036, 74186, 76188, 76699, 76706, 77742, 78636, 78958, 79814, 80584, 82516, 83818, 84242. 15 of these are redundant; (1596, 9581, 57491) ,(1834, 11009, 66059), (3589, 21539), (4396, 26381), (6872, 41237), (6986, 41921 ), (8258, 49553), (8743, 52463), (9366, 56201), (9576, 57461), (10694, 64169), (11142, 66857 ), (11389, 68339)
leaving 80 candidates for smallest seed.
1596, 1834, 3589, 4396, 6872, 6986, 8258, 8743, 9366, 9576, 10694, 11142, 11389, 14502, 15064, 16828, 17106, 17458, 18284, 19388, 19467, 19906, 20683, 20713, 21798, 23791, 25051, 26899, 27447, 28098, 28539, 29846, 30086, 32667, 33626, 35007, 35964, 36771, 37294, 40656, 43629, 43993, 44324, 47388, 48384, 48949, 49462, 49791, 51016, 53093, 54957, 57022, 57561, 58576, 58756, 59094, 59568, 60648, 61894, 62377, 63028, 63798, 65002, 66038, 66302, 68857, 70166, 73036, 74186, 76188, 76699, 76706, 77742, 78636, 78958, 79814, 80584, 82516, 83818, 84242.
There are 34 candidate seed values less that 84686 remaining after checking to 12000 digits.
Most of these have been eliminated. Some of the prime are available at Gary Barnes's Riesel Conjectures page.
 value prime at composite to value prime at composite to value prime at composite to 1596 1210000 26899 4542 58756 CRUS 1834 5533 27447 2135 59094 171929 3589 9186 28098 2355 59568 5996 4396 4175 28539 2545 60648 3857 6872 2268 29846 141526 61894 6215 6986 5119 30086 5437 62377 2559 8258 CRUS 32667 2122 63028 7317 8743 5417 33626 CRUS 63798 5821 9366 7845 35007 CRUS 65002 8507 9576 121099 35964 CRUS 66038 CRUS 10694 CRUS 36771 1210000 66302 3204 11142 3744 37294 CRUS 68857 7982 11389 CRUS 40656 CRUS 70166 4272 14502 4680 43629 5720 73036 7676 15064 2117 43993 569498 74186 CRUS 16828 CRUS 44324 2451 76188 7865 17106 6528 47388 4246 76699 8395 17458 CRUS 48384 6653 76706 CRUS 18284 CRUS 48949 143236 77742 560745 19388 CRUS 49462 CRUS 78636 4160 19467 3529 49791 CRUS 78958 458114 19906 CRUS 51016 528803 79814 113777 20683 CRUS 53093 2844 80584 2201 20713 3979 54957 CRUS 82516 2517 21798 CRUS 57022 483561 83818 3197 23791 2054 57561 2439 84242 3540 25051 2842 58576 7185
Conjectures 'R Us has checked most of these. The prime values for greater than 12000 digits are from CRUS.

#### Base 7

##### Digit 1
sn7,1(76) uses a period 12 cover to show that all terms are composite.
 Residue 0 2 0 3 1 Period 2 3 3 4 12 Prime 2 3 19 5 13
Primes dividing s12A+r7,1(76) are
 r 0 1 2 3 4 5 6 7 8 9 10 11 prime 2 13 2 19 2 3 2 5 2 19 2 3
The list of primes is {2,13,2,19,2,3,2,5,2,19,2,3}.
The smaller seed values produce primes
 Seed First Prime Seed First Prime Seed First Prime Seed First Prime Seed First Prime 1 2 3 3 4 4 1 5 18 6 1 7 4 8 3 9 2 10 1 11 2 12 127 13 424 14 5 15 1 16 1 17 20 18 3 19 256 20 4 21 4 22 3 23 468 24 5 25 10 26 9 27 14 28 1 29 2 30 1 31 4 32 27 33 4 34 1 35 2 36 17 37 4 38 5 39 40 40 1 41 2 42 31 43 4 44 3 45 2 46 45 47 2 48 1 49 6 50 3 51 260 52 5907 53 8 54 1 55 46 56 177 57 2 58 177 59 18 60 1 61 15118 62 3 63 16 64 1 65 938 66 1 67 4 68 15 69 2 70 1 71 8 72 7 73 4 74 3 75 398
##### Digit 2
sn7,2(15979) uses a period 12 cover to show that all terms are composite.
 Residue 1 0 1 2 11 Period 3 3 4 6 12 Prime 3 19 5 43 13
Primes dividing s30A+r7,2(15979) are
 r 0 1 2 3 4 5 6 7 8 9 10 11 prime 19 3 43 19 3 5 19 3 43 19 3 13
The list of primes is {19,3,43,19,3,5,19,3,43,19,3,13}.

31 candidates through 3000 digits.
(2047, 14331), (2261, 15829), 3601, 3667, 4011, 4101, 4315, 4785, 5149, 5507, 5849, 6479, 6609, 7505, 8305, 8911, 8937, 9069, 9331, 9839, 10279, 11837, 12347, 12565, 12979, 14331, 14759, 14945, 15181, 15711,.
Removing redundancies leaves 29 candidates less than 28521 that are checked to 3000 digits. 2047, 2261, 3601, 3667, 4011, 4101, 4315, 4785, 5149, 5507, 5849, 6479, 6609, 7505, 8305, 8911, 8937, 9069, 9331, 9839, 10279, 11837, 12347, 12565, 12979, 14759, 14945, 15181, 15711.
There are 17 candidate seed values less that 28521 remaining after checking to 15000 digits.
13096
 value prime at composite to value prime at composite to value prime at composite to 2047 15000 5849 6084 10279 3074 2261 6479 15000 11837 14782 3601 15000 6609 15000 12347 15000 3667 15000 7505 15000 12565 3159 4011 15000 8305 15000 12979 4836 4101 15000 8911 3839 14759 15000 4315 5502 8937 3242 14945 15000 4785 15000 9069 9263 15181 15000 5149 15000 9331 15000 15711 7429 5507 12597 9839 15000
##### Digit 3
sn7,3(5629) uses a period 12 cover to show that all terms are composite.
 Residue 0 1 2 2 0 Period 2 3 4 6 12 Prime 2 19 5 43 13
Primes dividing s12A+r7,3(5629) are
 r 0 1 2 3 4 5 6 7 8 9 10 11 prime 13 2 5 2 19 2 5 2 43 2 5 2
The list of primes is {13,2,5,2,19,2,5,2,43,2,5,2}.

54 candidates less than 5629 checked through 2000 digits.
(98,689,4826) (214, 1501) (358,2509), (515,3608), (556,3895), (589,4126) (734,5141), 896, 905, 989, 1126, 1246, 1336, 1516, 1639, 1796, 1945,2168, 2323, 2465, 2468, 2695, 2848, 2885, 3028, 3056, 3245, 3416, 3518, 3628, 3875, 4145, 4171, 4256, 4258, 4565, 4579, 4586, 4736, 4828, 4856, 5095, 5116, 5366, 5510, 5546, 5629
There are 46 candidates for smallest seed below 5629 checked through 2000 digits.
98, 214, 358, 515, 556, 589, 734, 896, 905, 989, 1126, 1246, 1336, 1516, 1639, 1796, 1945, 2168, 2323, 2465, 2468, 2695, 2848, 2885, 3028, 3056, 3245, 3416, 3518, 3628, 3875, 4145, 4171, 4256, 4258, 4565, 4579, 4586, 4736, 4828, 4856, 5095, 5116, 5366, 5510, 5546.
raising the check to 15000 digits leaves 19 smaller seed candidates.
 value prime at composite to value prime at composite to value prime at composite to 98 15000 1945 15000 4145 14802 214 3815 2168 15000 4171 8992 358 15000 2323 2964 4256 15000 515 15000 2465 15000 4258 3885 556 4895 2468 15000 4565 15000 589 15000 2695 4882 4579 15000 734 4669 2848 5589 4586 15000 896 5839 2885 15000 4736 2931 905 15000 3028 15000 4828 15000 989 15000 3056 2139 4856 3503 1126 15000 3245 15000 5095 15000 1246 2567 3416 3955 5116 15000 1336 15000 3518 15000 5366 4583 1516 2819 3628 15000 5510 5911 1639 2024 3875 2894 5546 15000 1796 3307
##### Digit 4
sn7,4(20277) uses a period 12 cover to show that all terms are composite.
 Residue 0 2 2 1 5 Period 3 3 4 6 12 Prime 3 19 5 43 13
Primes dividing s12A+r7,4(20277) are
 r 0 1 2 3 4 5 6 7 8 9 10 11 prime 3 43 19 3 13 19 3 43 19 3 5 19
The list of primes is {3,43,19,3,31,19,3,43,19,3,5,19}.

39 candidates checked through 2000 digits.
(507,3553),(2217,15523),(2303,16125), 3059,3405,3775,4203,4469,4515,4915,6517,6739,8687,10397, 10697,10813,12087,12293,12635,13095,13361,14583,14615,14801,14953,15599, 16211,16511,17313,18107,18139,18145,18863,19675,19703,20229,20277
There are 36 candidates less than 20277 checked through 2000 digits.
507,2217,2303,3059,3405,3775,4203,4469,4515,4915, 6517,6739,8687,10397,10697,10813,12087,12293,12635,13095, 13361,14583,14615,14801,14953,15599,16211,16511,17313,18107, 18139,18145,18863,19675,19703,20229.
Checking through 15000 digits leave 16 smaller seed candidates.
 value prime at composite to value prime at composite to value prime at composite to 507 15000 8687 15000 14953 15000 2217 3344 10397 15000 15599 4526 2303 2535 10697 15000 16211 3891 3059 3077 10813 13434 16511 15000 3405 15000 12087 6805 17313 4210 3775 4449 12293 15000 18107 15000 4203 15000 12635 15000 18139 10842 4469 15000 13095 2011 18145 9631 4515 2245 13361 2841 18863 2247 4915 7446 14583 2098 19675 15000 6517 15000 14615 2949 19703 15000 6739 15000 14801 3461 20229 2426
##### Digit 5
sn7,5(43) uses a period 6 cover to show that all terms are composite.
 Residue 0 0 1 5 Period 2 3 3 6 Prime 2 3 19 43
Primes dividing s12A+r7,5(43) are
 r 0 1 2 3 4 5 prime 2 19 2 3 2 43
The list of primes is {2,19,2,3,2,43}.

 Seed First Prime Seed First Prime Seed First Prime Seed First Prime 1 2 2 1 3 4 4 3 6 1 7 2 8 1 9 4 11 4 12 1 13 2 14 1 16 5 17 10 18 1 19 2 21 2 22 5 23 4 24 1 26 3 27 14 28 9 29 6 31 2 32 1 33 2 34 5 36 1 37 6 38 1 39 2 41 10 42 7 43 never 44 1

##### Digit 6
2384410917857141 uses a period of cover 60 sn7,6(2384410917857141) uses a period 60 cover to show that all terms are composite.
 Residue 2 3 0 4 2 6 9 1 9 18 48 Period 3 4 5 6 10 10 12 12 15 20 60 Prime 19 5 2801 43 11>/TD> 191 13 181 31 281 61
The primes dividing 60A + r are
 r 0 1 2 3 4 5 6 7 8 9 prime 2801 181 19 5 43 19 191 5 19 13 r 10 11 12 13 14 15 16 17 18 19 prime 2801 19 11 181 19 5 43 19 281 5 r 20 21 22 23 24 25 26 27 28 29 prime 19 13 11 19 31 2801 19 5 43 19 r 30 31 32 33 34 35 36 37 38 39 prime 2801 5 19 13 43 19 191 181 19 5 r 40 41 42 43 44 45 46 47 48 49 prime 2801 19 11 5 19 2801 43 19 61 181 r 50 51 52 53 54 55 56 57 58 59 prime 19 5 43 19 31 5 19 13 43 19
There are 4 values up to 105 checked through 3000 digits.
48251, 48583, 78647, 92017 Need to check 12m+(0;2;5;6;9), 12m+(5;8;11), 12m+(3;4;10),12m+(4; 7;8; 10;11), resp.

#### Base 8

##### Digit 1
sn8,1(21) uses a period 12 cover to show that all terms are composite.
 Residue 0 2 2 Period 2 4 4 Prime 3 5 13
Primes dividing s12A+r8,1(21) are
 r 0 1 2 3 prime 3 13 3 5
The list of primes is {3,13,3,5}.

 Seed First Prime Seed First Prime Seed First Prime Seed First Prime 1 2 2 1 3 3 4 12 5 1 6 21 7 2 8 2 9 1 10 6 11 1 12 1 13 314 14 1 15 3 16 2 17 1 18 139 19 20 20 2
##### Digit 3
sn8,3(1079770) uses a period 120 cover to show that all terms are composite.
 Residue 1 0 3 0 5 1 7 2 Period 3 4 4 6 8 8 10 12 Prime 73 5 13 19 17 241 11 37
Primes dividing s120A+r8,3(1079770) are
 r 0 1 2 3 4 5 6 7 8 9 prime 13 41 7 13 11 73 13 5 7 13 r 10 11 12 13 14 15 16 17 18 19 prime 271 5 13 4561 11 13 31 41 13 5 r 20 21 22 23 24 25 26 27 28 29 prime 7 13 61 5 13 41 7 13 4561 11 r 30 31 32 33 34 35 36 37 38 39 prime 13 5 7 13 11 5 13 1181 7 13 r 40 41 42 43 44 45 46 47 48 49 prime 271 41 13 5 11 13 31 5 13 11 r 50 51 52 53 54 55 56 57 58 59 prime 7 13 61 73 13 5 7 13 4561 5 r 60 61 62 63 64 65 66 67 68 69 prime 13 6481 7 13 11 41 13 5 7 13 r 70 71 72 73 74 75 76 77 78 79 prime 271 5 13 41 11 13 31 73 13 5 r 80 81 82 83 84 85 86 87 88 89 prime 7 13 61 5 13 6481 7 13 4561 11 r 90 91 92 93 94 95 96 97 98 99 prime 13 5 7 13 11 5 13 41 7 13 r 100 101 102 103 104 105 106 107 108 109 prime 271 73 13 5 11 13 31 5 13 11 r 110 111 112 113 114 115 116 117 118 119 prime 7 13 61 41 11 5 7 13 4561 5
The list of primes is {3,43,19,3,31,19,3,43,19,3,5,19}.

There are 721 candidates less than 1079770 listed at Base8Digit3.html checked to 15000 digits.
##### Digit 5
sn8,5(7476) uses a period 8 cover to show that all terms are composite.
 Residue 0 1 3 7 Period 2 4 8 8 Prime 3 13 17 241
Primes dividing s8A+r8,5(7476) are
 r 0 1 2 3 4 5 6 7 prime 3 13 3 17 3 13 3 241
The list of primes is {3,13,3,17,3,13,3,241}.

There are 29 candidates for minimum seed less than 7476 checked to 15000 digits.
 value prime at composite to value prime at composite to value prime at composite to 1241 15000 3458 15000 5703 7675 1673 15000 3503 15000 6267 15000 1791 15000 3762 8529 6312 15000 1898 15000 3801 15000 6611 15000 2046 15000 3843 7659 6851 15000 2171 15000 (486,3893) 15000 6896 13840 2274 15000 4121 15000 7014 15000 2336 15000 4283 15000 7232 15000 2489 15000 (539,4317) 15000 7233 15000 2553 8449 4464 15000 7272 7665 2723 11488 5058 7295 7398 15000 2982 15000 5292 15000 2999 7828 5531 15000 3116 10796 5627 15000
##### Digit 7
sn8,7(13) uses a period 4 cover to show that all terms are composite.
 Residue 0 1 3 Period 2 4 4 Prime 3 5 13
Primes dividing s4A+r8,7(13) are
 r 0 1 2 3 prime 3 5 3 13
The list of primes is {3,5,3,13}.

 Seed First Prime Seed First Prime Seed First Prime Seed First Prime 1 2 2 1 3 1 4 4 5 1 6 3 8 1 9 1 10 18 11 3 12 1

#### Base 9

##### Digit 1
111...1 = (9k-1)/8 = (3k-1)(3k+1)/(2*4)
##### Digit 2
sn9,2(4615) uses a period 6 cover to show that all terms are composite.
 Residue 0 2 0 1 Period 2 3 3 6 Prime 5 7 13 73
Primes dividing s8A+r9,2(4615) are
 r 0 1 2 3 4 5 prime 5 73 5 13 5 7
The list of primes is {5,73,5,13,5,7}.

The 14 candidates less than 4615 have been checked to 15000 digits.
 value prime at composite to value prime at composite to 307 15000 847 15000 1945 15000 975 15000 2555 6185 1125 15000 2995 15000 1157 15000 3157 15000 1197 7024 3437 9434 1255 15000 3985 15000 1637 11336 4157 5262 1679 15000 4167 15000 1787 15000 4195 15000
##### Digit 4
sn9,4(6059) uses a period 6 cover to show that all terms are composite.
 Residue 1 2 1 0 Period 2 3 3 6 Prime 5 7 13 73
Primes dividing s8A+r9,4(6059) are
 r 0 1 2 3 4 5 prime 73 5 7 5 13 5
The list of primes is {73,5,7,5,13,5}.

The 12 remaining candidate seeds less than 6059 checked to 18000 digits.
 value prime at composite to value prime at composite to 915 18000 3755 13011 1495 10423 3813 18000 1565 18000 3969 18000 1635 2479 3975 2041 2419 18000 4029 18000 2877 18000 4225 12379 2905 18000 4905 18000 3059 14290 5325 18000 3393 18000 5383 18000 3689 8428 5535 2045 5625 12157
Most of these are covered in the Base 3 digit 1 case. So the searches can be combined.
##### Digit 5
sn9,5(78) uses a period 6 cover to show that all terms are composite.
 Residue 0 1 0 5 Period 2 3 3 6 Prime 2 7 13 73
Primes dividing s8A+r9,5(78) are
 r 0 1 2 3 4 5 prime 2 7 2 13 2 73
The list of primes is {2,7,2,13,2,73}.

 Seed First Prime Seed First Prime Seed First Prime Seed First Prime 1 2 2 1 3 2 4 1 6 1 7 2 8 3 9 4 11 2 12 1 13 2 14 1 16 1 17 2 18 1 19 442 21 14 22 3 23 2 24 5 26 1 27 2 28 1 29 2 31 4 32 1 33 4 34 1 36 3 37 264 38 1 39 2 41 2 42 1 43 2 44 1 46 1 47 10 48 3 49 2 51 12 52 5 53 6 54 1 56 1 57 282 58 3 59 4 61 4 62 1 63 2 64 3 66 1 67 2 68 1 69 2 71 2 72 1 73 10 74 3 76 21 77 2
##### Digit 7
sn9,7(2) uses a period 2 cover to show that all terms are composite.
 Residue 1 0 Period 2 2 Prime 2 5
Primes dividing s2A+r9,7(2) are
 r 0 1 prime 5 2
The list of primes is {5,2}.

1779= 15110 is prime.
##### Digit 8
sn9,8(3)= (4*9k-1)= (2*3k-1)*(2*3k+1).
189 = 1710 is prime.

#### Base 10

##### Digit 1
Lenny Jones proved that 37 is the minimal seed and produced the table showing the first appearance of a prime for each smaller seed.
sn10,1(37) uses a period 6 cover to show that all terms are composite.
 Residue 0 1 0 5 Period 2 3 3 6 Prime 2 7 13 73
Primes dividing s8A+r10,1(37) are
 r 0 1 2 3 4 5 prime 7 3 37 13 3 37
The list of primes is {7,3,37,13,3,37}.

##### Digit 3
The only candidate seed less that 4070 is 817. This has been checked to 554789 digits.
Candidates requiring at least 2500 digits to produce a prime are
 seed digits 410 37398 817 1037 3292 1166 12689 1279 4752 2959 6763 3674 16097
##### Digit 7
The minimal seed is 891.
##### Digit 9
Two candidates less than 10175 remain.
 value prime at composite to 449 11958 1343 29711 1802 45882 1934 51836 3355 13323 4015 3647 4420 630000 4477 4817 6587 5846 6664 60248 7018 630000 8578 373260

H. Riesel, Några stora primtal (Swedish: Some large primes), Elementa 39 (1956), 258-260.
L. Jones, When does appending the same digit repeatedly on the right of a positive integer generate a sequence of composite numbers?, Amer. Math. Monthly 118 (2011), 153-160.
The Reisel problem
The repunit case for other bases OEIS A084740
Karsten Bonath's Riesel Prime database No Prime left behind summary A condensed table for the extended Riesel conjectures at No Prime Left Behind