## The curious powers of 1 + sqrt 2: conjecture

In my previous post I related the following puzzle from Colin Wright:

What’s the 99th digit to the right of the decimal point in the decimal expansion of $(1 + \sqrt 2)^{500}$?

Let’s play around with this a bit and see if we notice any patterns. First, $1 + \sqrt 2$ itself is approximately

$1 + \sqrt 2 \approx 2.414213562373095\dots$

so its powers are going to get large. Let’s use a computer to find the first ten or so:

$\displaystyle \begin{array}{cc} n & (1 + \sqrt 2)^n \\ \hline 1 & 2.414213562373095 \\ 2 & 5.82842712474619 \\ 3 & 14.071067811865474 \\ 4 & 33.970562748477136 \\ 5 & 82.01219330881975 \\ 6 & 197.99494936611663 \\ 7 & 478.002092041053 \\ 8 & 1153.9991334482227 \\ 9 & 2786.0003589374983 \\ 10 & 6725.9998513232185 \end{array}$

Sure enough, these are getting big (the tenth power is already bigger than $6000$), but look what’s happening to the part after the decimal: curiously it seems that the powers of $(1 + \sqrt 2)$ are getting rather close to being integers! For example, $(1 + \sqrt 2)^{10}$ is just under $6726$, only about $0.0002$ away.

At this point, I had seen enough to notice and conjecture the following patterns (and I hope you have too):

• The powers of $(1 + \sqrt 2)$ seem to be getting closer and closer to integers.
• In particular, they seem to alternate between being just under an integer (for even powers) and just over an integer (for odd powers).

If this is true, the decimal expansion of $(1 + \sqrt 2)^{500}$ must be of the form $n.99999999\dots$ for some big integer $n$ and some number of $9$s after the decimal point. And it seems reasonable that if Colin is posing this question, it must have more than 99 nines, which means the answer would be 9.

But why does this happen? Do the powers really keep alternating being just over and under an integer? And how close do they get—how do we know for sure that $(1 + \sqrt 2)^{500}$ is close enough to an integer that the 99th digit will be a 9? This is what I want to explore in a series of future posts—and as should come as no surprise it will take us on a tour of some fascinating mathematics!