Problems & Puzzles: Puzzles

Puzzle 575.- Primes | k leftmost digits are divided by p(k)

Here I'm asking by the smallest prime p with d digits such that the k leftmost digits are divided by p(k), for k=1 to d-1.

Here are the first smallest prime-cases:

 d p 2 23 3 211 4 2153 5 21567 6 240131 7 2401367 8 24508901 9 420095519 10 ???

Contributions came from Giovanni Resta, Jan van Delden, Seiji Tomita & Farideh Firoozbakht, Torbjörn Alm, J. K. Andersen, Hakan Summakoğlu, Igor Schein, Emmanuel Vantieghem, Alexandre Patarot.

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Almost all of the puzzlers discovered the same: a) my table is wrong b) the solution for d=10 and c) no more solutions after d=10.

23
3 211
4 2153
5 21001
6 210761
7 2156053
8 21076943
9 215605927
10 6650672641

There are no further terms.

There are no more terms, because the requirement that the resulting prime should have d digits exactly prohibits other solutions.

If one keeps track of a list of the possible first d-1 digits and extend this list in the next step we eventually run into the problem that the divisibility criterion of the next prime will shorten the length of this list (to zero). Or to put it differently, the new prime will force the resulting numbers in the new list to have more digits than allowed.

There is only another possible prime for d=10, i.e., 6650672647.

we ask for divisibility for k, then we have the following table
of primes:
2 11
3 101
4 1021
5 10243
6 102001
7 1020001
8 10200611
9 102054409
10 1024023241
11 10200616207
12 102006882013
13 1020061620607
14 10200616206043
15 102006162060463
16 1020061620604657
17 10885280406000001
18 105258321084840029
19 1264025610120000967
20 12600072001272009613
21 144408645048225636601
22 2408588820001056602071
23 74125808108436004200067
24 123606009012225672009013  (unique)
25 none

Did you notice that your solution p=21567 for d=5 is not a prime?

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Farideh wrote:

a(10) = 6650672641 and a(11) doesn't exist, namely there is no 11-digit prime with the mentioned property.

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Emmanuel wrote:

First I determined all numbers  m  of which the  k  leftmost digits are divisible by  p(k)  for  k = 1, 2, ... , d, where  d  is the number of digits of  m :
2, 4, 6, 8, 21, 24, 27, 42, 45, 48, 60, 63, 66, 69, 81, 84, 87, 210, 215, 240, 245, 270, 275, 420, 425, 450, 455, 480, 485, 600, 605, 630, 635, 660, 665, 690, 695, 810, 815, 840, 845, 870, 875, 2100, 2107, 2156, 2401, 2408, 2450, 2457, 2702, 2709, 2751, 2758, 4200, 4207, 4256, 4501, 4508, 4550, 4557, 4802, 4809, 4851, 4858, 6006, 6055, 6300, 6307, 6356, 6601, 6608, 6650, 6657, 6902, 6909, 6951, 6958, 8106, 8155, 8400, 8407, 8456, 8701, 8708, 8750, 8757, 21076, 21560, 24013, 24508, 24574, 27027, 27093, 27511, 27588, 42009, 42075, 45012, 45089, 45507, 45573, 48026, 48092, 48510, 48587, 60060, 60555, 63008, 63074, 63569, 66011, 66088, 66506, 66572, 69025, 69091, 69586, 81554, 84007, 84073, 84568, 87010, 87087, 87505, 87571, 210769, 215605, 240136, 245089, 270270, 270933, 275119, 275886, 420095, 420758, 450125, 450892, 455078, 480922, 485108, 485875, 600600, 605553, 630084, 630747, 660114, 660881, 665067, 690911, 695864, 815542, 840073, 840736, 845689, 870103, 870870, 875056, 875719, 2107694, 2156059, 2401369, 2450890, 2751195, 2758862, 4200955, 4207585, 4508927, 4550781, 4851086, 4858753, 6055536, 6300846, 6307476, 6608818, 6650672, 6958644, 8155427, 8400737, 8407367, 8708709, 8750563, 8757193, 21076947, 21560592, 27588627, 42009551, 45507812, 63008465, 66506726, 81554270, 84007379, 84073670, 87571931, 210769470, 630084655, 665067264, 875719319, 6300846559, 8757193191
See A079206 of the OEIS.
Then it was not difficult to adjust and complete the table :
d  p
2  23
3  211
4  2153
5  21001
6  210761
7  2156053
8  21076943
9  215605927
10 6650672641
There is no  p  for  d > 10.

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Alexandre Patarot wrote:

For now, the only results I have are the following one:
23;
211;
2153;
21001;
210761;
2156053;
21076943;

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