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Problems & Puzzles: Puzzles

Puzzle 21.- Happy primes

If you iterate the process of summing the square of the decimal digits of a number, then it’s easy to see that you either reach the cycle

4à 16à 37à 58à 89à 145à 42à 20à 4 or arrive at 1. In the latter case you started from a "happy number".

a) Find the least "happy prime" of k digits, for 1 <= k <= 10.

b) Can you give an arithmetic caracterization of the "happy numbers", that is to say, can you predict somehow when a given number is a "happy number"?

This type of numbers were created by Reg Allenby’s daugther (p. 234 Ref. 2)

Solution

Harvey Heinz and Jud McCranie obtained independently (19/09/98) the following solutions for k=1 to 15:

7
13
103
1009
10009
100003
1000003
10000121
100000039
1000000009
10000000033
100000000003
1000000000039
10000000000411
100000000000067

Jud obtained the following others for k=16 to 19

1000000000000487
10000000000000481
100000000000000003
1000000000000000003

all of them are happy primes and the least for each size.

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In September of 2004, Joseph Galante calculated more terms, now up to k=50:

10000000000000000000000000000000000000000000000009 (50)

Then on my request he calculated the earliest titanic: 10^999 +663

***

J. K.Andersen wrote:

http://www.worldofnumbers.com/em144.htm  shows the smallest gigantic
probable primes. The first happy is 10^9999+70999.

Milton L. Brown found the probable prime 10^100000+342909 in 2007.
It is happy and I assume it is the smallest with 100001 digits.

With help from the GMP library I computed decimal expansions of the
largest known primes in the Prime Pages database on 31 October 2008.
The top-10 happy primes are listed below.
"all" is the position among all known primes.
"digits" is the number of decimal digits.

# all happy prime digits iterated sum of squares of digits
-- -- ------------------- ------ ---------------------------------
1 22 4847*2^3321063+1 999744 28495502->219->86->100->1
2 24 3*2^3136255-1 944108 26897464->302->13->10->1
3 28 7*2^2915954+1 877791 25000461->82->68->100->1
4 38 1183953*2^2367907-1 712818 20292693->219->86->100->1
5 45 5077*2^2198565-1 661838 18920431->176->86->100->1
6 50 23*2^2141626-1 644696 18372557->226->44->32->13->10->1
7 55 251749*2^2013995-1 606279 17301182->129->86->100->1
8 56 467917*2^1993429-1 600088 17094438->236->49->97->130->10->1
9 59 25*2^1977369-1 595249 16949498->376->94->97->130->10->1
10 60 121*2^1954243-1 588288 16788102->219->86->100->1

4847*2^3321063+1 was found by Richard Hassler in the Seventeen or Bust
project in 2005. None of the 21 larger known primes are happy.
It appears from http://www.shaunspiller.com/happynumbers  that around
1/7 of all numbers are happy, so none in the top-20 primes is unlucky.

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