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
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Sources

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