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Is 5 a happy number?

Published in Number Theory 1 min read

No, 5 is not a happy number.

Understanding Happy Numbers

A happy number is a number which eventually reaches 1 when replaced by the sum of the square of each digit. If this process leads into a cycle that does not include 1, the number is an unhappy number.

Determining if 5 is a Happy Number

Let's follow the process for the number 5:

  1. 52 = 25
  2. 22 + 52 = 4 + 25 = 29
  3. 22 + 92 = 4 + 81 = 85
  4. 82 + 52 = 64 + 25 = 89
  5. 82 + 92 = 64 + 81 = 145
  6. 12 + 42 + 52 = 1 + 16 + 25 = 42
  7. 42 + 22 = 16 + 4 = 20
  8. 22 + 02 = 4 + 0 = 4
  9. 42 = 16
  10. 12 + 62 = 1 + 36 = 37
  11. 32 + 72 = 9 + 49 = 58
  12. 52 + 82 = 25 + 64 = 89

Notice that we have encountered 89 again. This indicates that we have entered a cycle that will never reach 1. Therefore, 5 is not a happy number; it is an unhappy number. The numbers 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, 16, 17, 18, 20, ... are all unhappy numbers.

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