Problems & Puzzles: Puzzles

 Puzzle 808. Palprimes that remain... Dmitry Kamenetsky sent the following nice puzzle: Here we ask for palprimes base 10 that remain palindrome in several other bases. Example: 191, base=10 -> 515, base=6 and 232, base=9 Simplifying: 191(10) -> 515(6), 232(9) Q. Find other palprimes base 10 that remain palindrome in  more bases than 191.

Contributions came from Emmanuel Vantighem, Jan van Delden, T. D, Noe, Ramón David Aznar

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Emmanuel wrote:

There are many palprimes that remain palprime in other bases than 10.
Here I a made a record list(*) of the palprime  p (< 3*10^10) followed by the set  S  of all bases (less than p) for which  p  is palprime :
151,  {3, 10}
191,  {6, 9, 10}
373,  {4, 8, 9, 10}
10301,  {10, 27, 30, 44, 100}
18481,  {10, 105, 110, 112, 120, 132}
19891,  {10, 31, 32, 61, 102, 117, 130}
96769,  {10, 47, 67, 209, 224, 252, 256, 288}
1008001,  {10, 119, 221, 326, 720, 750, 800, 840, 875, 896, 900, 960, 1000}
1062601,  {10, 160, 173, 255, 422, 489, 759, 770, 805, 825, 840, 920, 924, 966, 1012}
1670761,  {10, 121, 161, 348, 591, 918, 936, 945, 952, 1020, 1071, 1080, 1092, 1105, 1170, 1190, 1224, 1260}
101474101,  {10, 824, 1000, 1619, 1983, 6029, 7350, 7434, 7644, 7670, 7965, 8190, 8260, 8673, 8775, 8820, 8850, 9100, 9204, 9450, 9555, 9828}
10008180001,  {10, 3329, 12715, 70980, 71487, 72800, 73125, 73320, 74025, 76050, 76375, 76986, 78000, 78750, 78960, 79430, 81120, 81900, 82250, 84000, 84500, 84600, 85176, 85540, 87984, 88125, 88725, 90000, 91000, 91650, 93600, 94000, 94640, 94752, 95316, 97500, 97760, 98700}
12137973121,  {10, 3022, 3651, 15684, 37033, 56629, 78120, 78435, 78624, 79680, 80352, 82336, 83328, 83664, 84240, 84630, 86320, 87048, 87360, 87399, 87885, 89280, 89640, 90055, 90272, 90636, 90720, 92628, 92960, 93744, 94122, 95616, 96720, 97110, 98280, 100347, 100440, 101556, 102920, 103584, 104160, 104580, 104832, 107136, 107568, 108066, 108810}
14063936041,  {10, 3339, 5350, 9961, 84645, 85140, 85785, 86184, 86526, 86940, 87032, 87780, 88236, 89010, 89397, 89870, 91080, 91770, 92340, 93555, 93955, 94392, 95634, 96140, 97524, 97911, 98040, 99330, 100947, 101574, 102168, 102465, 102942, 103845, 104328, 104490, 105336, 106191, 106260, 106812, 106920, 107730, 107844, 108360, 108790, 110124, 110295, 111780, 112746, 112860, 114380, 114939, 115368, 117990, 118503}
1000605060001,  {10, 100, 50257, 709800, 711200, 714375, 721500, 733044, 739375, 740000, 742950, 750360, 751205, 751840, 757575, 760500, 762000, 772668, 777000, 780000, 781625, 787878, 789432, 792480, 794131, 800100, 808080, 819000, 822325, 825500, 832104, 832500, 841750, 845000, 845820, 851760, 855218, 858520, 865800, 866775, 875420, 887250, 889000, 900432, 901446, 901875, 910000, 914400, 916305, 924560, 932400, 937950, 939800, 946400, 950625, 952500, 962000, 965835, 969696, 971250, 977392, 986790, 990600, 1000125}
1002113112001,   {10, 20191, 36897, 112606, 708000, 708708, 715080, 722150, 722750, 726880, 728000, 735280, 736320, 742350, 744875, 746592, 750750, 751660, 763224, 764400, 767000, 772044, 774670, 777700, 778800, 786588, 787800, 791840, 795025, 800800, 805350, 808000, 808500, 808808, 809952, 816585, 826000, 826826, 833250, 834260, 840840, 843700, 848400, 852137, 858000, 859040, 862400, 866580, 867300, 871024, 872256, 875875, 875973, 882336, 885885, 888800, 893850, 901992, 908600, 917686, 919100, 920400, 924000, 924352, 925120, 929604, 933240, 939575, 944944, 950208, 953440, 954030, 955500, 960960, 965055, 972125, 973500, 983235, 984750, 989800, 991200, 1001000}  (82 bases)

(*) That  p, S  figures in this list means that for any palprime q < p  the number of bases for which  q  remains palprime is strictly less than the number of elements in  S

I also found only two palprimes below 10^15 that remain palprime in base 2 :
313 =  (100111001)(2)
7284717174827 = (1101010000000011010111110101100000000101011)(2)

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Jan wrote:

I searched all palprimes p with a maximum of 17 digits.

Longest sequence of representations in bases b, b<10:

373(10): 454(9),565(8),11311(4)

I found no other palprime p with more than 1 other representation.
I found no palprime p with a representation in base 5.

Representation in other bases b, b>10:

If  b>p we have the trivial solution p(b).
Every representation, except the trivial representation [1,1](p-1), consists of an odd number of digits in base b:

If we have a representation with an even number of digits in base b:  [a[0],a[1],..,a[n-1],a[n-1],...,a[1],a[0]](b) we have p=a[0]*(1+b^(2n-1))+a[1]*b*(1+b^(2n-3))+..+a[n-1]*b^(n-1)*(1+b).
Since b+1 is a divisor of b^(2k-1)+1, p has a divisor b+1. So p can only be prime if p=b+1 (or b=p-1 having representation [1,1](b)).

Representations with an odd number of digits 2k+1, k>0, have necessarily   p^(1/(2k+1))<b<p^(1/(2k)). Strict inequalities since p is prime.

If p^(1/3)<b<p^(1/2) we have three digits. The representation is of the form [a,c,a](b). Hence p-a=b(ab+c). If a=1 and p-1 has many prime factors, we expect many solutions.
For instance: p=14063936041 with 14063936040=2^3*3^5*5*7*11*19*23*43 has 51 representations of the form [1,c,1](b) and 3 of the form [a,c,a] with a>1.

Some interesting representations where all digits and the base are prime:

155292551(10): [3,5,13,11,13,5,3](19)

1551551(10):     [11,17,3,17,11](19)
15551(10):         [11,13,11](37) =[2,67,2](73) This one is for Sheldon:

"The best number is 73. Why? 73 is the 21st prime number. Its mirror, 37, is the 12th and its mirror, 21, is the product of multiplying 7 and 3... and in binary 73 is a palindrome, 1001001, which backwards is 1001001."

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Tony wrote:

How appropriate that the number of this puzzle - 808 - is also a palindrome!

If one works with just the bases 2 to 10, then it appears that 373 is the
base-10 palindrome that has the most bases that also produce palindromes:
4, 8, 9, and 10.  We have 373(10) = 454(9) = 565(8) = 11311(4).

However, a number can be palindromic in bases higher than 10.  As an
extreme example, every number n > 1 is palindromic in base n-1.  Here is a
list of numbers palindromic in base 10 that have an increasing number of
palindromes in other bases:

5, {2,4}

151, {3,10,150}

191, {6,9,10,190}

373, {4,8,9,10,372}

10301, {10,27,30,44,100,10300}

18481, {10,105,110,112,120,132,18480}

19891, {10,31,32,61,102,117,130,19890}

96769, {10,47,67,209,224,252,256,288,96768}

1008001, {10,119,221,326,720,750,800,840,875,896,900,960,1000,1008000}

1062601, {10,160,173,255,422,489,759,770,805,825,840,920,924,966,1012,1062600}

1670761,
{10,121,161,348,591,918,936,945,952,1020,1071,1080,1092,1105,1170,1190,1224,1260,1670760}

There appears to be no limit to the number of such palindromes.

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Ramón wrote:

```nº bases	Palprime	Bases
1	11	10
2	151	3	10
3	191	6	9	10
4	373	4	8	9	10
5	10301	10	27	30	44	100
6	18481	10	105	110	112	120	132
7	19891	10	31	32	61	102	117	130
8	96769	10	47	67	209	224	252	256	288
13	1008001	10	119	221	326	720	750	800	840	875	896	900	960	1000
15	1062601	10	160	173	255	422	489	759	770	805	825	840	920	924	966	1012
18	1670761	10	121	161	348	591	918	936	945	952	1020	1071	1080	1092	1105	1170	1190	1224	1260
22	101474101	10	824	1000	1619	1983	6029	7350	7434	7644	7670	7965	8190	8260	8673	8775	8820	8850	9100	9204	9450	9555	9828```

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Hakan Summakoglu wrote:

373(10) -> (11311)4, (565)8, (454)9

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