MathOverflow

In this question on MSE, Enzo Creti asks for a prime number formed by concatenating the Mersenne numbers $2^n-1$ and $2^{n-1}-1$, for example, $40952047$. For all residues modulo $7$, he found primes except for the residue $6$. This is somewhat surprising...

Background: My daughter is 6 years old now, once I wanted to think on some math (about some Young diagrams), but she wanted to play with me... How to make both of us to do what they want ? I guess for everybody who has children, that question comes up...

Claim: I present a proof of global regularity for the 3D incompressible Navier-Stokes equations with arbitrary large H¹ initial data.
Main result: For divergence-free $u_0 \in H^1(\mathbb{R}^3)$, the unique Leray-Hopf weak solution satisfies$$\|u(t)\|_{L^2}^2 \leq C (1+t)^{-\delta}$$for some $\delta > 0$, with $C$ depending only on $\|u_0\|_{H^1}$. This energy decay rules out finite...

Let $\Omega^+$ be a finite polygonal domain in $\mathbb{P}^1(\mathbb{C})$, by which we mean that its boundary is a polygon with only finite points.
Let$$ \Omega^- = \mathbb{P}^1(\mathbb{C}) \setminus \overline{\Omega^+}.$$
Let $(f, g)$ be a pair of functions, where $f$, resp. $g$, is a continuous complex-valued function on $\overline{\Omega^+}$, resp. $\overline{\Omega^-}$, which is holomorphic on $\Omega^+$, resp. $\Omega^-$. Denote by $S$ the space of such pairs of functions...