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Happy Numbers

Rumman Ansari November 16, 2024 599 views

Definition

A number \(n\) is called a Happy Number if, starting with \(n\), repeatedly replacing the number by the sum of the squares of its digits, the process eventually leads to the number 1. If the process results in an infinite loop of numbers that does not include 1, then \(n\) is considered an Unhappy Number.

Mathematical Process

Let \(S(n)\) represent the sum of the squares of the digits of a number \(n\). The process is as follows:

\[ n_1 = S(n), \quad n_2 = S(n_1), \quad n_3 = S(n_2), \quad \dots \]

If for some \(n_k = 1\), then \(n\) is a Happy Number. Otherwise, the number falls into a cycle and is considered Unhappy.

Example

Consider the number 19:

\[ 1^2 + 9^2 = 1 + 81 = 82 \] \[ 8^2 + 2^2 = 64 + 4 = 68 \] \[ 6^2 + 8^2 = 36 + 64 = 100 \] \[ 1^2 + 0^2 + 0^2 = 1 \]

Since the process ends at 1, 19 is a Happy Number.

Mathematical Notation of the Process

The general formula for checking if a number \(n\) is Happy can be written as:

\[ \text{If } S^k(n) = 1 \text{ for some } k \geq 1, \text{ then } n \text{ is a Happy Number.} \]

Conclusion

If the iterative process eventually reaches \(1\), the number is classified as a Happy Number. If it falls into a cycle that doesn't reach 1, it's considered an Unhappy Number.