MathOverflow

I am investigating the $p$-adic analytic continuation of the discrete sum $\sum_{i=1}^n i^d \lfloor i^{p^k}/p^k \rfloor$. By writing the sum as $\sum_{m = 0}^n a_m\binom{n}{m}$, we obtain its explicit Mahler expansion.
I am trying to find the exact number of $p$-adic roots in $\mathbb{Z}_p$ for the resulting continuous function:...

I am an independent researcher. This arose in the context of studying the Beal conjecture.Setup: Factor A³+B³=(A+B)(A²-AB+B²)=Cⁿ. For coprime A,B: gcd(A+B, A²-AB+B²) divides 3. This gives two cases.
Case 1: gcd(A+B, A²-AB+B²)=1Write A+B=sⁿ, A²-AB+B²=tⁿ. Then AB=(s²ⁿ-tⁿ)/3 and the discriminant condition gives:4tⁿ - s²ⁿ = 3u²for some positive integer u (required for A,B to be positive integers).The equation 4tⁿ-s²ⁿ=3u² can be reformulated as a norm...

Note that I’ve asked this question on mathstackexchange last month. I am looking for book recommendations on complex dynamics that include discussion of polynomial mating.
Ideally, the book would cover foundational material (Julia sets, the Mandelbrot set, quadratic polynomials, etc.) and introduce or at least meaningfully discuss polynomial mating. I have a background with complex analysis and some familiarity with dynamical...

Does the category $\mathbf{FreeAb}$ of free abelian groups have sequential colimits? I assume that the answer is No, but what is an explicit sequence of free abelian groups and homomorphisms$$A_0 \to A_1 \to A_2 \to \cdots$$which has no colimit in $...