MathOverflow

I have the following questions about light condensed mathematics. In the non-light setting, the corresponding statements are addressed in Scholze’s lecture notes and in Mann’s thesis, but I have not been able to find a reference in the light case.
(1) For any light profinite set $S$, is $\mathbb{Z}[S]$ projective in $\mathrm{Cond}(\mathrm{Ab})^{\mathrm{light}}$?...

The Banach space $\ell_\infty$ is 1-injective so it is 1-complemeneted in any superspace. Let $P$ be a contractive projection from $L_\infty$ onto an isometric copy of $\ell_\infty$. Can $P$ be arranged in a way that $\| I - P\| < 2$?
If so, this would improve the recent Banach-Mazur distance bound between the two spaces found by Plebanek and Korpalski....

Motivation: As a personal side project I have been working with an inclusion-exclusion formulation that is counting weighted power’s $x^a$ between consecutive squares $[n^2, (n+1)^2]$. The function $f(x)$ however involves a lot of floor functions and...

Landau's fourth problem conjectures that there are infinitely many primes of the form $a²+1$. I have been looking at the relationship between these primes and the progression $4n+1$.
If $a$ is odd, $a = 2m+1$. Then $a²+1 = (2m+1)²+1 = 4m²+4m+2 = 2(2m²+2m+1)$. Thus, for $a > 1$, $a²+1$ is always even and not prime....