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.
*** 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 top10 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*2^31362551 944108 26897464>302>13>10>1
3 28 7*2^2915954+1 877791 25000461>82>68>100>1
4 38 1183953*2^23679071 712818 20292693>219>86>100>1
5 45 5077*2^21985651 661838 18920431>176>86>100>1
6 50 23*2^21416261 644696 18372557>226>44>32>13>10>1
7 55 251749*2^20139951 606279 17301182>129>86>100>1
8 56 467917*2^19934291 600088 17094438>236>49>97>130>10>1
9 59 25*2^19773691 595249 16949498>376>94>97>130>10>1
10 60 121*2^19542431 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 top20 primes is unlucky.
***
