Problems & Puzzles: Puzzles

Puzzle 843. Looking for more Kaprekar prime numbers Our past puzzles 837 and 838 were devoted to Kaprakar prime numbers. In the first of these ( 837 )we announced that R(19) was the first and only Kaprekar prime number known, and invited to found more. Emmanuel Vantieghem and Jan van Delden shown that every R(n) such that n is prime of the form 9k+1 was Kaprekar and primes. Accordingly R(109287) and R(270343) must be also Kaprekar prime numbers. In the second of these ( 838 ) we invited to find more prime numbers out of the Repunits, without giving any kind of clue. The result was that we had not any success. Even, we received not any contribution. Now we provide a clue, hoping this to be a good clue. The prime Kaprekar solutions already found (the three repunits already mentioned above) are part of the following infinite family of Kaprekar numbers: R(n), for n=9k+1. Are there more infinity families of Kaprekar numbers such that at the same they are odds and not ending in 5? Again, by inspection of the first 51514 Kaprekar numbers computed by R. Gerbicz and  given here, http://oeis.org/A006886/b006886.txt, we found the following three of such families:   No Infinity family if Kaprekar integers, odd not ending in 5, k=>0 1 (6)k7(0)k+1(3)k+1 2 6(1)9k+306(1)9k+4 3 3(8)9k+33(8)9k+29 Q1. Can you find a prime or a probable prime in some the these infinite family of integers? Q2. Can you find other infinite families promissory for our target?

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Contribution came from Emmanuel Vantieghem

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Emmanuel wrote:

 Q1.

Family 1 : 

If we set  10^k = x, then every element of the family can be written as

  (2 x^3 + x^2 + x - 1)/3 = (2 x - 1)((x^2 + x + 1)/3)

and so, no element is prime.

 

Family 2 :

If we set  10^(9k+4) = x, then every element of the family can be written as

  (550 x^2 - 45 x - 1)/9 = (55 x + 1)((10 x - 1)/9),

hence never prime.

 

Family 3 :

If we set  10^(9k+3) = x, then every element of the family can be written as

  (350 x^2 - 45 x + 1)/9 = (35 x - 1)((10 x - 1)/9)

which is never prime.

 

All elements of the three families are Kaprekar numbers.
 

Q2. I could not find any other family of Kaprekar numbers wich could deliver candidates for prime Kaprekar numbers.

P.S. R19 is still the onliest prime among the first 11th million Kaprekar numbers (seen at http://chesswanks.com/seq/b006886/).   

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