Birthday Special: Happy Numbers

It’s my 20th birthday today.
To celebrate, here’s a special blog post about a fun concept in number theory: happy numbers.

In number theory, a happy number is a number which eventually reaches 1 when the number is replaced by the sum of the squares of each digit.
— Wikipedia

Let’s take a look at two numbers in particular.

20 (the age I am now) is a sad number :(.

Proof

$20 \Rightarrow 2^2 + 0^2 = 4$
$4 \Rightarrow 4^2 = 16$
$16 \Rightarrow 1^2 + 6^2 = 1 + 36 = 37$
$37 \Rightarrow 3^2 + 7^2 = 9 + 49 = 58$
$58 \Rightarrow 5^2 + 8^2 = 25 + 64 = 89$
$89 \Rightarrow 8^2 + 9^2 = 64 + 81 = 145$
$145 \Rightarrow 1^2 + 4^2 + 5^2 = 1 + 16 + 25 = 42$
$42 \Rightarrow 4^2 + 2^2 = 16 + 4 = 20$ — we have reached a loop
$20 \Rightarrow \dots$ and it begins again

Thus, the squared sum of the digits of 20 reaches an infinite loop and never terminates by reaching 1, implying that 20 is a sad number. This concludes the proof.

But there is hope for me.

2026 is a happy number :).

Proof

$2026 \Rightarrow 2^2 + 0^2 + 2^2 + 6^2 = 4 + 0 + 4 + 36 = 44$
$44 \Rightarrow 4^2 + 4^2 = 16 + 16 = 32$
$32 \Rightarrow 3^2 + 2^2 = 9 + 4 = 13$
$13 \Rightarrow 1^2 + 3^2 = 1 + 9 = 10$
$10 \Rightarrow 1^2 + 0^2 = 1$ — we have reached 1

Thus, the squared sum of the digits of 2026 eventually reaches 1, implying that 2026 is a happy number. This concludes the proof.

That’s all. Happy birthday to my February 16th twins!