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?

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged , . Bookmark the permalink.

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

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