Happy Number - Happy Primes

Happy Primes

A happy prime is a number that is both happy and prime. The happy primes below 500 are

7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487 (sequence A035497 in OEIS).

All numbers, and therefore all primes, of the form 10n + 3 and 10n + 9 for n greater than 0 are Happy (This does not mean that these are the only happy primes, as evidenced by the sequence above). To see this, note that

• All such numbers will have at least 2 digits;
• The first digit will always be 1 due to the 10n
• The last digit will always be either 3 or 9.
• Any other digits will always be 0 (and therefore will not contribute to the sum of squares of the digits).
  • The sequence for adding 3 is: 12 + 32 = 10 → 12 = 1
  • The sequence for adding 9 is: 12 + 92 = 82 → 82 + 22 = 64 + 4 = 68 → 62 + 82 = 36 + 64 = 100 -> 1

The palindromic prime 10150006 + 7426247×1075,000 + 1 is also a happy prime with 150,007 digits because the many 0's do not contribute to the sum of squared digits, and, which is a happy number. Paul Jobling discovered the prime in 2005.

As of 2010, the largest known happy prime is (Mersenne prime). Its decimal expansion has 12,837,064 digits.