A happy prime is a number that is both a happy number and a prime number.
Understanding Happy Primes
To fully understand what a happy prime is, let's break down the two concepts that define it:
-
Happy Number: A happy number is a number that, when you repeatedly replace it by the sum of the squares of its digits, eventually reaches 1. If the process loops endlessly in a cycle not including 1, then the number is an unhappy number.
-
Prime Number: A prime number is a whole number greater than 1 that has only two divisors: 1 and itself.
Definition of Happy Prime
A happy prime combines these two properties. Thus, a happy prime is a number that satisfies both the conditions of being a happy number and being a prime number.
Example
The reference gives us the example of 19. 19 is a 10-happy prime.
Let's verify this:
- Is 19 a Happy Number?
- 12 + 92 = 1 + 81 = 82
- 82 + 22 = 64 + 4 = 68
- 62 + 82 = 36 + 64 = 100
- 12 + 02 + 02 = 1 + 0 + 0 = 1
Since 19 eventually reaches 1 through this process, it's a happy number.
- Is 19 a Prime Number?
19 is only divisible by 1 and 19, so it is a prime number.
Since 19 is both happy and prime, it's a happy prime.
Important Note
Unlike happy numbers in general, rearranging the digits of a happy prime does not guarantee another happy prime. The reference specifically states that while 19 is a 10-happy prime, 91 (which is a rearrangement of the digits of 19) is not a prime number because 91 = 13 × 7. However, 91 is still a happy number.
