Triple of Consecutive Happy Numbers

Theorem

The smallest triple of consecutive integers all of which are happy is:

$\left({1880, 1881, 1882}\right)$

Proof

\(\ds \)

\(\)

\(\ds 1880\)

\(\ds \leadsto \ \ \)

\(\ds 1^2 + 8^2 + 8^2 + 0^2\)

\(=\)

\(\ds 1 + 64 + 64 + 0\)

\(\ds \)

\(=\)

\(\ds 129\)

\(\ds \leadsto \ \ \)

\(\ds 1^2 + 2^2 + 9^2\)

\(=\)

\(\ds 1 + 4 + 81\)

\(\ds \)

\(=\)

\(\ds 86\)

\(\ds \leadsto \ \ \)

\(\ds 8^2 + 6^2\)

\(=\)

\(\ds 64 + 36\)

\(\ds \)

\(=\)

\(\ds 100\)

\(\ds \leadsto \ \ \)

\(\ds 1^2 + 0^2 + 0^2\)

\(=\)

\(\ds 1 + 0 + 0\)

\(\ds \)

\(=\)

\(\ds 1\)

and so $1880$ is happy

\(\ds \)

\(\)

\(\ds 1881\)

\(\ds \leadsto \ \ \)

\(\ds 1^2 + 8^2 + 8^2 + 1^2\)

\(=\)

\(\ds 1 + 64 + 64 + 1\)

\(\ds \)

\(=\)

\(\ds 130\)

\(\ds \leadsto \ \ \)

\(\ds 1^2 + 3^2 + 0^2\)

\(=\)

\(\ds 1 + 9 + 0\)

\(\ds \)

\(=\)

\(\ds 10\)

\(\ds \leadsto \ \ \)

\(\ds 1^2 + 0^2\)

\(=\)

\(\ds 1 + 0\)

\(\ds \)

\(=\)

\(\ds 1\)

and so $1881$ is happy

\(\ds \)

\(\)

\(\ds 1882\)

\(\ds \leadsto \ \ \)

\(\ds 1^2 + 8^2 + 8^2 + 2^2\)

\(=\)

\(\ds 1 + 64 + 64 + 4\)

\(\ds \)

\(=\)

\(\ds 133\)

\(\ds \leadsto \ \ \)

\(\ds 1^2 + 3^2 + 3^2\)

\(=\)

\(\ds 1 + 9 + 9\)

\(\ds \)

\(=\)

\(\ds 19\)

\(\ds \leadsto \ \ \)

\(\ds 1^2 + 9^2\)

\(=\)

\(\ds 1 + 81\)

\(\ds \)

\(=\)

\(\ds 82\)

\(\ds \leadsto \ \ \)

\(\ds 8^2 + 2^2\)

\(=\)

\(\ds 64 + 4\)

\(\ds \)

\(=\)

\(\ds 68\)

\(\ds \leadsto \ \ \)

\(\ds 6^2 + 8^2\)

\(=\)

\(\ds 36 + 64\)

\(\ds \)

\(=\)

\(\ds 100\)

\(\ds \leadsto \ \ \)

\(\ds 1^2 + 0^2 + 0^2\)

\(=\)

\(\ds 1 + 0 + 0\)

\(\ds \)

\(=\)

\(\ds 1\)

and so $1882$ is happy

This theorem requires a proof. In particular: It remains to be shown that this is the smallest such triple You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1880$