Question 74
A positive integer is called happy if repeatedly replacing it by the sum of the squares of its digits eventually gives $1$. For example, $1900$ is happy because
$$
1900 \rightarrow 82 \rightarrow 68 \rightarrow 100 \rightarrow 1
$$
a) Find two consecutive two-digit happy numbers.
b) Prove that for every $n\gt 2020$ there are two consecutive happy numbers with $n$ digits.
$$
1900 \rightarrow 82 \rightarrow 68 \rightarrow 100 \rightarrow 1
$$
a) Find two consecutive two-digit happy numbers.
b) Prove that for every $n\gt 2020$ there are two consecutive happy numbers with $n$ digits.