Infinite series squared question.

Question

kainui · Answer

Is it possible to only pick integer coefficients $a_n$ on a sum between 0 and 4 inclusive to get a solution to:

$$\left(\sum_{n=0}^\infty \frac{a_n}{5^n} \right)^2 = 2$$

watchmath · Answer

yes, just write $\sqrt{2}$ down in base 5 ?

kainui · Answer

Haha yep.

kainui · Answer

I guess I didn't really think this through, cause in my mind I was going to say there were 2 solutions, and then present this other one using the $a_n$ from the previous solution to make this one, although this one requires infinitely many digits to the right and I think is not entirely "finished" since you can carry...

$$\sum_{n=1}^\infty 4*5^n+ \sum_{n=-\infty}^0 (5-a_n)5^n$$

I think I'll just leave it and move on since I need to think about this more; this is like playing around with 5-adic numbers.