(1) A disjunctive word theorem for an integer sequence.
Does your phone number show up somewhere as a subword of some prime number? How about any arbitrary finite sequence of digits, does there exist a prime that contain it as a subword? What about the sequence given by $a_n = 3n^4 + 2n^3 - 4n +17$?
(2) Density and powers of two.
Are the power of two encyclopedic? That is, is your phone number also in some power of two? This time the argument will take us looking at the irrationality of some number and how multiples of them fill up an interval densely.
Commuting polynomial chain.
What can we say about commuting polynomials, under composition? A chain of polynomial is a set of polynomials with exactly one polynomial per degree $d\ge 0$. What are all the polynomial chains that is commutative? Hints. Note we only need to consider up to conjugation by an invertible linear poylnomial. So without loss the degree 2 member of a commuting polynomial chain is of the form $x^2 + k$, for some $k$. Then consider the conditions of $p_2$ commuting with $p_3$ to show that $k=0$ or $-2$.
Ritt's polynomial decomposition theorem.
Given a polynomial $p(x)$ that has degree 2 or more. If there exists some $\phi_1(x),\phi_2(x)$ each with degree 2 or more, such that $p(x)=(\phi_1\circ\phi_2)(x)$, then we say this is a decomposition. A polynomial $p(x)$ is said to be prime if it does not admit any decompositions. As it turns out, if a polynomial $p$ admits prime decomposition $p=\phi_1\circ\phi_2\circ\cdots\circ\phi_k$, then the number of prime factors and their degrees are unique, up to reordering. This reminds me of Jordon-Holder theorem.
