Proof if the digits of the golden ratio are in sequence in the digits of Pi or not

Question

Since the digits of $\pi$ are uniformly random and infinite, any finite sequence of digits can be found in sequence somewhere among the digits of $\pi$.

But does this also holds when it comes to finding the digits of another irrational number like $\phi$ (the golden ratio) in $\pi$?

My intuition is that the digits of $\phi$ would not occur in sequence in the digits of $\pi$. This intuition is probably mostly based on the pigeon hole theory, but that only applies to discrete numbers and infinity tends to make things weird.

Is there some proof to show that the digits of $\phi$ (or any another irrational number) does or does not occur in sequence in the digits of $\pi$?

Edit to add to clarity: Does the decimal expansion of $\pi$ become the expansion of $\phi$ for the remaining (infinitely many) integers, as in, say, $\pi$=3.1415⋯1618033⋯?

Answer 1

As mentioned in the comments, this would contradict the transcendence of $\pi$.

To see this, note that $$\Phi=\frac {1+\sqrt 5}2=1.618033\cdots $$ is algebraic.

Thus if we could write $\pi$ as $$\pi=3.\underbrace {141592\cdots}_{n\,\text{terms}}1618033\cdots $$

we could conclude that $10^{n+1}\pi = N+\Phi$ for some $N\in \mathbb N$ and this would imply that $\pi$ were algebraic, yielding a contradiction.

Note that this argument does not work if, say, you replace $\Phi$ with some transcendental irrational, such as $e^{100}$. Such cases would need to depend on the precise number you chose and the techniques involved would be considerably more difficult.

Answer 2

You appear to be referring to the property of normal numbers. In a normal number, any finite sequence of digits will appear and, more specifically, with a particular frequency.

However, note two important points.

1. It is only speculated that $\pi$ is normal. (See the link above)

2. Since $\phi$ is irrational, its digits are not finite so even if $\pi$ were proven to be normal, it still might not be the case that it contains the digits of $\phi$.

In fact, it almost certain that the digits of $\phi$ do not appear in $\pi$ but it depends a bit on what you mean. In what sense might one infinite sequence be within another? If it means that at some point, the digits of $\pi$ would those of match $\phi$ then the answer is no as lulu explains in a comment; this would mean that $\pi$ was also algebraic which we know that it isn't. If you mean something like "every second digit of $\pi$ would match $\phi$ then maybe but it is not guaranteed by the normality of $\pi$.