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
Jud obtained the following others for k=16 to 19 1000000000000487
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 ***
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