Problems & Puzzles: Puzzles

 Puzzle 989. Smallest integer that generate at least n different primes Paolo Lava sent the following nice puzzle:   Take an integer k, insert it into itself and divide this number by k getting a rational x. Of course, if k has j digits you can perform j-1 divisions generating j-1 values for x. My question is: which is the smallest integer that generate at least n different primes? I made a search and here are the first three results: n = 1, k = 11, x = 101  because 1111/11 = 101. n = 2, k = 185, x = 641 and 983  because 118585/185 = 641 and 181855/185 = 983. n = 3, k = 213675, x = 1036001, 992801 and 999749 because 221367513675/213675 = 1036001, 212136753675/213675 = 992801, 213213675675/213675 = 997841 (not prime), 213621367575/213675 = 999749, 213672136755/213675 = 4999933/5 (obviously not prime). Q. Can we find additional solutions for n = 4, 5, 6, etc. ?

Along the week 8 to 14 February 2020, contributions came from Paul Leary, Giovanni Resta, Emmanuel Vantieghem, Oscar Volpatti

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Paul wrote:

Here is the first solution I found for n = 4.

2710027100 Has 4 Primes {8380000001, 10000900001, 9999983801, 10000000631},

From

{22710027100710027100, 27102710027100027100, 27100227100271007100, 27100272710027100100}

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Giovanni wrote:

The smallest number that gives 4 primes is 2710027100. The multiples and
the primes are:

22710027100710027100 => 8380000001,
27100227100271007100 => 9999983801,
27100272710027100100 => 10000000631,
27102710027100027100 => 10000900001.

I searched for the smallest number that gives 5 primes up to 7.5*10^12
without success,

however, one such number, probably not minimal, is 1307189542483660 :

13013071895424836607189542483660 => 9955000000000001,
13071130718954248366089542483660 => 9999415000000001,
13071895413071895424836602483660 => 9999999991000001,
13071895424813071895424836603660 => 9999999999982001,
13071895424836130718954248366060 => 9999999999999641.

I found some other numbers that give 5 primes:

3267973856209150, 9803921568627450, 107105370263265000,
142807160351020000, 178508950438775000, 285614320702040000, etc.

One that gives 6 primes: 209414435355857950000.
(The primes are 965044000000000000001, 999999931168000000001,
999999984424600000001, 999999999982027000001, 999999999999996463541,
1000031068000000000001).

And one number that gives 8 primes: 231589170445738750000.
(The primes are 963598240000000000001, 998455967200000000001,
999999795815920000001,
999999999902500480001, 999999999997810084001, 999999999999932677841,
999999999999972217577, 1000000264016800000001).

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Emmanuel wrote:

The number  2710027100   generates four primes : 10000000631, 9999983801, 10000900001, 8380000001
It is the smallest (if I made no mistakes).
The number  1307189542483660  generates five primes : 9999999999999641, 9999999999982001, 9999999991000001, 9999415000000001, 9955000000000001
But I'm not sure it is the smallest.

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Oscar wrote:

Dear Carlos,
I'll submit my findings with k<1e29 (my current searching algorithm runs out of memory for 30-digit numbers).
p_m is the prime obtained inserting a copy of k after the m least significant digits of k itself

n = 4;
k = 2710027100 (10 digits);
p_3 = 10000000631;
p_4 =   9999983801;
p_6 = 10000900001;
p_9 =   8380000001;

n = 5;
k = 1307189542483660 (16 digits);
p_2  = 9999999999999641;
p_4  = 9999999999982001;
p_7  = 9999999991000001;
p_11 = 9999415000000001;
p_13 = 9955000000000001;

n = 6;
k = 209414435355857950000 (21 digits);
p_6  =   999999999999996463541;
p_10 =   999999999982027000001;
p_13 =   999999984424600000001;
p_14 =   999999931168000000001;
p_17 = 1000031068000000000001;
p_19 =   965044000000000000001;

n = 8;
k = 231589170445738750000 (21 digits);
p_7  =   999999999999972217577;
p_8  =   999999999999932677841;
p_9  =   999999999997810084001;
p_11 =   999999999902500480001;
p_14 =   999999795815920000001;
p_15 = 1000000264016800000001;
p_18 =   998455967200000000001;
p_20 =   963598240000000000001;

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