My motivation for this question is to establish a systematic approach to solving problems of this type. Specifically, there are numerous questions on Math.StackExchange.com that focus on numbers and their digits. Here are some examples:
Problems about digits of powers of two:
Is $2048$ the highest power of $2$ with all even digits (base ten)?
Status of a conjecture about powers of 2
Is $2^{16} = 65536$ the only power of $2$ that has no digit which is a power of $2$ in base-$10$?
Problems about decimal digits of irrational numbers:
Constructing number between zero and one by concatenating digits from square root of primes
Number made from the first digits of $2^n$
Swapping the digits of an algebraic number (e.g. $\sqrt 2$)
For fixed positive integer $k$, do we have a digit in $\sqrt{2}$ that repeats every $k$ digits?
https://mathoverflow.net/q/265310/173402
https://youtu.be/bG7cCXqcJag?si=Xq-oVb_nSnlCKw42
Kindly share your knowledge to help answer this question and contribute to the advancement of mathematics.
