You'll find something interesting if you take the golden ratio, , to high powers:
It seems as if large exponents of come arbitrarily close to integer values. In fact, isn't the only number which does this. For example, . This is no coincidence.
These numbers - known as Pisot-Vijayaraghavan numbers - produce "almost-integers" when raised to high powers. I'll give a quick sketch of why this is, as well as an easy way to find other numbers with this property:
We can take a number , where and are integers, and consider how it behaves under an operation called Galois conjugation. Essentially, this is just a more general version of what you may already know as complex conjugation. To find the Galois conjugate of , simply change the sign of the square root term: . This is equivalent to setting to be the root of the polynomial and finding the second root, or .
A few properties of Galois conjugation can be easily deduced, and are already implicit in the quadratic equation above:
The last equality is important here. Notice especially that these four equations all produce integers results on the righthand side. Now, whenever , the second term of becomes arbitrarily small for increasing . This immediately implies that
Thus, any number whose Galois conjugate satisfies will produce near-integer values when raised to high powers. Here are some examples of such numbers:
*It's worth noting that in the case that has a denominator of , as is the case with where , then we also need the further restriction that , or that is divisible by so that remains an integer.