Problems & Puzzles: Puzzles

 Puzzle 837. Kaprekar prime numbers On June 26, 2016, G. L. Honaker asked to me by email for the minimal Kaprekar prime number, if it existed. According to the Wikipedia: "...a Kaprekar number for a given base is a non-negative integer, the representation of whose square in that base can be split into two parts that add up to the original number again. For instance, 45 is a Kaprekar number, because 452 = 2025 and 20+25 = 45. The Kaprekar numbers are named after D. R. Kaprekar." A few minutes ago I discovered the minimal prime Kaprekar number analyzing in Excel the list the first 51514 Kaprekar numbers computed by R. Gerbicz and  given here, http://oeis.org/A006886/b006886.txt : 1111111111111111111 (the 764th Kaprekar number listed) BTW, 1111111111111111111 is the smallest prime Kaprekar number and also the second prime Repunit. 1111111111111111111 is a Kaprekar number because 1111111111111111111^2 = 1234567901234567900987654320987654321 and 123456790123456790 + 0987654320987654321 = 1111111111111111111 Q. Find three more Kaprekar prime numbers.

Contributions came from Jan van Delden, Emmanuel Vantieghem and Shyam S. Gupta.

Jan wrote:

Given:

X^2=A*N+B
X=A+B
0<B<N

[Which we will use with N=10^k for some k]

Theorem by Ianucci (2000):

A number X is a Kaprekar number if and only if X=d*m  with m=Inv(d,(N-1)/d), with Inv(a,b) the multiplicative inverse of a mod b and d an unitary divisor of N-1.

[Source: Wikipedia]

Given (a,b)=1: Inv(a,b)=m is defined by the smallest value 0<m<b such that a*m=1 mod b.

We search X=d*m, with X prime and N=10^k.

We must have that either:

d=1 and m=p (prime) or

m=1 and d=p (prime).

d=1: m=Inv(1,N-1) or m=1 mod (N-1), so m=1 giving X=1 (not prime)

d=p: m=1=Inv(p,(N-1)/p) or p=1 mod ((N-1)/p)

Since p and (N-1)/p are odd we must have that p=(2t)(N-1)/p+1 (t>=1) or p>2(N-1)/p or p^2>2(N-1)>N-1.

Thus p must be the largest prime divisor of N-1 having at least half the number of digits of 2(N-1).

Write N-1=9*R[k], with R[k] the repunit having k digits.

Write R[k]=3^a*B[k] where a is the multiplicity of the factor 3 in R[k]

Since p cannot be 3 (it is not a unitary divisor of N-1):

p=1 mod ((N-1)/p) or  p=1 mod (3^(2+a) * B[k]/p) or

p=1 mod (3^(2+a)) and

p=1 mod (B[k]/p)

If R[k] is prime, choose p=R[k], we have a=0 and B[k]=R[k] and get R[k]=1 mod 9 or k=1 mod 9.

The first prime solution has k=19.
The other known (probable) prime repunits having this property are:

R[109297] probably prime (Harvey Dubner, april 3 2007)

R[270343] probably prime (Maksym Voznyy, july 15 2007)

The next one (when found) has k>2500000.

[Source: sequence A004023 OEIS]

If R[k] is not prime I used the factorisations of R[k] given by Yousuke Koide and checked k<=250.

I found no other solutions.

In short, prime Kaprekar may be other than certain prime repunits?... Possibly.

***

Emmanuel wrote:

1) If  k  is a Kaprekar number we have :
k = a+b  and  k^2 = b + a 10^n.
Modulo 9 this leads to :
k^2 = k (mod 9)  or  k(k-1) = 0 (mod 9).
If in addition  k  is prime (and thus > 3), then it follows  k = 1 (mod 9).

2) On the other hand, it is a well known property that any Kaprekar number  k  can be written as a product
d * u
where  d  is a unitary divisor of  N = 10^n - 1 (that is a divisor of  N  that has no divisor in common with  N/d)
and  u  is the smallest among all positive numbers  x  that have the property  d*x = 1 (mod N/d).
The converse is also true.

3) Suppose  r  is a repunit with  s  ones, s = 1 (mod 9).
Thus : r = (10^s - 1)/9, i.e. : r  is a divisor of  S = (10^s - 1)
and it is also a unitary divisor of  S, since  S/r = 9  has no divisor in common with  r.
Furthermore, we have :
r = 1 (mod S/s).
Hence, r  is a Kaprekar number with  u = 1.

Conclusion : when a repunit which is = 1 (mod 9)  is prime; then it is a Kaprekar number.
Unfortunately there is only one proved prime repunit : R(19).
According to   A004023, the other candidates are  R(109297)  and  R(270343)  which are probable primes.

Of course, there might exist prime divisors  d  of  N =10^n - 1  that are not repunits but that satisfy  d = 1 (mod N/d).
These numbers would be Kaprekar numbers when they are unitary divisors.  But I could not find such one neither I could prove they do not exist

***

Shyam wrote:

I have been investigating this issue since last one year and my results are as under:

The Kaprekar numbers which are primes can be termed as Kaprekar Primes.
The smallest Kaprekar prime is 1111111111111111111 i.e. R19
On investigation of Kaprekar numbers I found that the next probable kaprekar primes are the following :
(10^109297-1)/9
and (10^270343-1)/9
these are two largest known repunit probable primes.

Based on the investigation of Kaprekar numbers I found that digital root of Kaprekar numbers can only
be 1 or 9 that is if Kaprekar number is denoted by K then K (mod 9) can be 1 or 9. So for Kaprekar
number to be prime, K (mod 9) must be 1.
On checking all repunit primes including probable primes I found that all Repunit primes(mod 9) which
are equal to 1 are all Kaprekar numbers.

Based on my investigations I conjecture that :
(i) A Kaprekar number can be prime only if it is Repunit prime with digital root as 1.
(ii) Any Repunit Prime with digital root as 1 is a Kaprekar prime number.

So following Known Repunit primes including probable primes are the only known Kaprekar primes.
(10^19-1)/9 (R19 prime)
, (10^109297-1)/9 ( R109297 probable prime)
and (10^270343-1)/9 (R270343 probable prime).

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