Problems & Puzzles: Puzzles

Puzzle 15.- Narcissistic and Handsome Primes

Michael Keith has used the word "narcissistic" for numbers like the 153, such that :

153 = 13 + 53 + 33

Switching to primes I ask :

a) find the three first "narcissistic prime numbers"?

Now let’s relax the conditions and let’s permit any integer power for each digit (not necessarily the same for different digits) and let’s name this numbers just "handsome" numbers.

b) find the least "handsome pal-primes" for 3, 5, 7, etc. digits.


Solution

Jo Yeong Uk, report us by e-mail (26/10/98) the following:

"There are 88 Narcissistic numbers proved by D.Winter in 1985 and verified by D.Hoey.You can see more information in below site. http://mathworld.wolfram.com/NarcissisticNumber.html 

I have factored all narcissistic numbers except trivial composite numbers, and I found 7 Narcissistic prime numbers:   2, 3, 5, 7, 28116440335967, 449177399146038697307, 35452590104031691935943"

This information solves completely the question a) of this puzzle: "find the three first "narcissistic prime numbers".

***

Lionel E. Deimel, Ph.D, from Pittsburgh, USA, wrote (13/06/2000) a very documented comment about these issues:

"Carlos, I am writing about various aspects of Puzzle 15 on your Web site. First, you speak of "narcissistic numbers" and refer to Mike Keith. The term actually seems to have originated with Joseph S. Madachy in his 1966 book "Mathematics on Vacation." He used it to refer to all numbers "that are representable, in some way, by mathematically manipulating the digits of the numbers themselves." He then goes on to discuss digital invariants, VR numbers, etc. Numbers such as 153, which you mention in conjunction with Puzzle 15, are "pluperfect digital invariants" (PPDIs), which are, of course, a kind of narcissistic number. See my Web site ( http://www.deimel.org ) for precise definitions of PPDIs and PDIs or "perfect digital invariants."

The PDI/PPDI terminology was coined by Max Rumney in a 1962 article in "Recreational Mathematics Magazine." Numbers such as 153 were first remarked upon by G.H. Hardy in his book "A Mathematician's Apology," but he gave no name to them and dismissed them as being of no real importance. Mike Keith, at his Web site, is mostly concerned with unusual narcissistic numbers, by the way ( http://members.aol.com/s6sj7gt/mikewild.htm  )

By asking for the "first three narcissistic prime numbers," you are actually asking for the first three base-10 PPDIs. (PDIs and PPDIs can be studied in different bases.) Your definition of "handsome" numbers I have not seen before. It is an interesting idea, though perhaps best called something like "mixed digital invariant," for consistency. The first nontrivial decimal prime of this type is 43 (= 4^2 + 3^3).

Jo Yeong Uk's report on your Web site is somewhat misleading. To begin with, the list of all PPDIs in base-10 (there are actually 89, not 88, assuming 0 is counted as a PPDI) was reported by Michael Jones and me in "Journal of Recreational Mathematics" in 1982. (It is on my Web site.) Dik Winter does have a list of these and other PPDIs on the Web ( http://www.cwi.nl/ftp/dik/Armstrong ), where he calls them Armstrong numbers. (I have seen this term often, but I have no idea where it came from.) Jo Yeong Uk exhibits only six prime PPDIs, of course, as 2 is not considered prime. Michael Jones and I noted in JRM that 28116440335967 is prime, and we didn't think it important to find any more base-10 PPDI primes. Regards, Lionel"

***

On May 5, 2005, Giovanni Resta wrote:

In Point (b) of Puzzle 15 you asked about handsome palindromic primes.
Here is a summary of my findings about handsome numbers.

In the following, I have assumed that 0^0=0 (while for x>0 x^0=1)

Regarding handsome palprimes (HPP) here is what I found:
There is only one HPP with 3 digits, and two representations:
373 = 3^1 + 7^3 + 3^3 = 3^4 + 7^2 + 3^5 And there is only one HPP with 5 digits:
98389 = 9^4 + 8^1 + 3^1 + 8^5 + 9^5

There are 26 7-digits HPP:
3223223 = 3^7 + 2^10 + 2^16 + 3^7 + 2^20 + 2^21 + 3^8
3233323 = 3^0 + 2^11 + 3^8 + 3^9 + 3^13 + 2^14 + 3^13
3245423 = 3^8 + 2^13 + 4^4 + 5^7 + 4^10 + 2^21 + 3^8 ............................
9749479 = 9^1 + 7^3 + 4^1 + 9^7 + 4^8 + 7^6 + 9^7
9795979 = 9^0 + 7^3 + 9^2 + 5^10 + 9^4 + 7^5 + 9^4
9852589 = 9^0 + 8^0 + 5^6 + 2^13 + 5^10 + 8^4 + 9^5

There are several 9-digits HPP. The first 3 and the last 3 are:
125838521 = 1^1 + 2^17 + 5^10 + 8^1 + 3^6 + 8^4 + 5^11 + 2^26 + 1^1
127636721 = 1^1 + 2^11 + 7^5 + 6^9 + 3^0 + 6^9 + 7^9 + 2^26 + 1^1
129535921 = 1^1 + 2^10 + 9^0 + 5^0 + 3^17 + 5^8 + 9^1 + 2^12 + 1^1 ...............................
982393289 = 9^3 + 8^4 + 2^18 + 3^11 + 9^8 + 3^15 + 2^29 + 8^6 + 9^9
972434279 = 9^2 + 7^6 + 2^25 + 4^3 + 3^15 + 4^6 + 2^29 + 7^6 + 9^9
976252679 = 9^5 + 7^0 + 6^6 + 2^19 + 5^11 + 2^29 + 6^8 + 7^7 + 9^9

Finally, the first 11-digit HPP is:
12236663221 = 1^1+2^9+2^23+3^19+6^0+6^2+6^8+3^21+2^26+2^29+1^1

Another class of interesting handsome numbers are those which are palindromic and have a representation where also the exponents are symmetric. Apart the trivial ones (n<10) the first ones are:
262 = 2^7 + 6^1 + 2^7
4224 = 4^3 + 2^11 + 2^11 + 4^3
32823 = 3^3 + 2^14 + 8^0 + 2^14 + 3^3
39393 = 3^9 + 9^1 + 3^2 + 9^1 + 3^9
79597 = 7^1 + 9^3 + 5^7 + 9^3 + 7^1
2232322 = 2^16 + 2^20 + 3^0 + 2^12 + 3^0 + 2^20 + 2^16

and a couple of larger examples are:

286343682 = 2^23 + 8^5 + 6^4 + 3^12 + 4^14 + 3^12 + 6^4 + 8^5 + 2^23
924292429 = 9^0 + 2^28 + 4^0 + 2^9 + 9^9 + 2^9 + 4^0 + 2^28 + 9^0

while the first (and only, up to 78427372487) such number which is also prime is

34763936743 = 3^14+4^16+7^0+6^13+3^6+9^8+3^6+6^13+7^0+4^16+3^14

For what concernes the pandigital handsome numbers, among the 10!-9!=3265920 pandigital 10-digits numbers, there are 192288 handsome numbers. The smallest and the largest are

1023456879 = 1^1 + 0^1 + 2^27 + 3^18 + 4^6 + 5^1 + 6^11 + 8^9 + 7^5 + 9^7
9876532041 = 9^0 + 8^11 + 7^9 + 6^9 + 5^13 + 3^15 + 2^16 + 0^1 + 4^10 + 1^1

***


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