Smallest Adjacent Happy Numbers
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Theorem
The smallest adjacent happy numbers are $31$ and $32$.
Proof
This can be determined by testing all the positive integers in succession for happiness.
Checking $31$ and $32$:
| \(\ds 31\) | \(\to\) | \(\ds 3^2 + 1^2 = 9 + 1 = 10\) | ||||||||||||
| \(\ds \) | \(\to\) | \(\ds 1^2 = 1\) | and so $31$ is happy. |
| \(\ds 32\) | \(\to\) | \(\ds 3^2 + 2^2 = 9 + 4 = 13\) | ||||||||||||
| \(\ds \) | \(\to\) | \(\ds 1^2 + 3^2 = 1 + 9 = 10\) | ||||||||||||
| \(\ds \) | \(\to\) | \(\ds 1^2 = 1\) | and so $32$ is happy. |
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $31$