## Puzzle | Kaprekar & Repunit numbers

Repunit    https://en.wikipedia.org/wiki/Repunit
$R_{n} \; = \; (10^n-1)/9$

Repunit primes are repunits that are prime.   For example,

$R_{2}$,     $R_{19}$,     $R_{23}$

The only other known repunit primes are the ones with 317 digits and with 1,031 digits:

$R_{317}$     and     $R_{1031}$

Kaprekar number
https://en.wikipedia.org/wiki/Kaprekar_number

$R_{19}$   is the smallest prime Kaprekar number

$1111111111111111111^2 = 1234567901234567900987654320987654321$

$123456790123456790 + 0987654320987654321 = 1111111111111111111$

Can you find more prime Kaprekar numbers?

math grad - Interest: Number theory
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### 4 Responses to Puzzle | Kaprekar & Repunit numbers

1. paul says:

I couldn’t find any repunit primes up to 10000 digits and no more repunit prime Kaprekar numbers up to 1031 digits. There are many Kaprekar numbers of which these repunits are Kaprekar numbers
these are the number of digits up to 1000 digits

{19,28,37,46,55,64,73,82,91,100,109,118,127,136,145,154,163,172,181,190,199,208,217,226,235,244,253,262,271,
280,289,298,307,316,325,334,343,352,361,370,379,388,397,406,415,424,433,442,451,460,469,478,487,496,505,
514,523,532,541,550,559,568,577,586,595,604,613,622,631,640,649,658,667,676,685,694,703,712,721,730,739,748,
757,766,775,784,793,802,811,820,829,838,847,856,865,874,883,892,901,910,919,928,937,946,955,964,973,982,
991,1000}

Paul.

2. pipo says:

I also couldn’t find any more repdigits 1 length 1 (mod 9) which are prime either.
But something interesting:
Repdigit 5 length 2 (mod 9) are also all Kaprekarnumbers, for example:
55*55 = 3025 => 30 + 25 = 55
55555555555*55555555555 = 3086419753024691358025 =>
30864197530 + 24691358025 = 5555555555.
Repdigit 7 length 4 (mod 9) are also all Kaprekarnumbers,
7777 * 7777 = 60481729 => 6048 + 1729 = 7777,
7777777777777^2 = 60493827160481728395061729 =>
6049382716048 + 1728395061729 = 7777777777777, etc.
Repdigit 2 length 5 (mod 9) also: 22222^2 = 493817284, 4938 + 17284 = 22222, 22222222222222^2 = 493827160493817283950617284 =>
4938271604938 + 17283950617284 = 22222222222222 etc.
Repdigit 4 length 7 (mod 9) also: 4444444^2 = 19753082469136, 1975308+2469136 = 44444444
4444444444444444^2 = 19753086419753082469135802469136 =>
1975308641975308 + 2469135802469136 = 4444444444444444, etc.
Repdigit 9, 99, 999, 9999, 99999, 999999 etc. are all Kaprekar numbers.

Repdigits 3 and 6 are not Kaprekar numbers.

pipo