Recall that I call $r>0$ a Galois radius of an integer $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ primes and positive $a$ and $b$ and a primality radius of $n$ if $a=b=1$.
Does it suffice to prove that the number $N_{Gal}(n)$ of Galois radii of $n$ is greater than $f(n):=\sum_{p\leq\sqrt{n}}\lfloor\frac{\log n}{\log p}\rfloor$ to deduce $n$ has at least a primality radius? If I'm not mistaken one has $f(n)=O(\sqrt{n})$.
Edit August 9th 2022: I've been told that one has $f(n)=O\left(\frac{\sqrt{n}}{\log n}\right)$ which, incidentally, is of the order of magnitude of the square root of the expected number of primality radii of $n$.
Edit August 15th 2022: I've been told that rewriting $f(n):=\sum_{p^{i}\leq n ,p\leq\sqrt{n},i\geq 1}1$ and setting $g(n):=\sum_{n\leq p^{i}\leq 2n,p\leq\sqrt{2n},i\geq 1}1$, it would suffice to prove that $f(n)+g(n)<N_{Gal}(n)$ to ensure the existence of a Galois radius $r$ of $n$ which would be a primality radius of $n$.
Note that these sums can be expressed generically as $f(n):=\sum_{p^{i}\in D,p\leq\partial^{+}D,i\geq 1}$ and $g(n):=\sum_{p^{i}\in D',p\leq\partial^{+}D',i\geq 1}1$ where $D=(0,n)$, $D'=(n,2n)$ and $\partial^{+}E:=\sqrt{\sup E}$, and that $D'$ is the image of $D$ by the involution $x\mapsto 2n-x$. That way, denoting by $S:=\sup\{\Re(s)\mid\zeta(s)=0\}$, do we have $\max(f(n),g(n))=O_{\varepsilon}(N_{Gal}(n)^{S+\varepsilon}\}$ and $\min(f(n),g(n))=O_{\varepsilon}(N_{Gal}(n)^{1-S+\varepsilon}\}$ (call this Zeta vanishing Growth conjecture, or ZVG conjecture for short)?
Edit August 16th 2022: a probabilistic approach might be possible. I read in a French math book about probability theory that given $n$ real independent random variables $X_{1},\cdots X_{n}$ and writing $W:=\min(X_{1},\cdots X_{n})$ and $Z:=\max(X_{1},\cdots X_{n})$, one has $F_{Z}=\prod_{i=1}^{n}F_{X_{i}}$ and $F_{W}=1-\prod_{i=1}^{n}(1-F_{X_{i}})$ where $F_{Y}$ is the cumulative distribution function of the random variable $Y$.
Writing symbolically $P$ for the product of the CDF and $\sigma$ for the involution $x\mapsto 1-x$, one can establish the following duality between $F_{Z}$ and $F_{W}$: $P^{\dagger}:=\sigma\circ P\circ \sigma^{-1}\Longrightarrow (P^{\dagger})^{\dagger}=P$.
My idea would be to apply this to random variables $X_{f}$ and $X_{g}$ of respective expectancy $f(n)$ and $g(n)$ so as to keep track of the duality between $D$ and $D'$.
Edit August 17th 2022: setting $\beta_{min}:=\inf\{\beta\mid\min(f(n),g(n))=O(N_{Gal}(n)^{\beta}\}$ and $\beta_{\max}:=\inf\{\beta\mid\max(f(n),g(n)=O(N_{Gal}(n)^{\beta}\}$, call the conjectured relation $\beta_{min}=\sigma(\beta_{max})$ the "min/max symmetry hypothesis". The Goldbach conjecture is equivalent to $f(n)\sim g(n)$ as it boils down to the sequence of Galois radii of $n$ to "tend" to an arithmetic progression (this the meaning of $\alpha_{n}=o(n)$ in my previous question About Goldbach's conjecture). That way we would have $\beta_{min}=\beta_{max}$ and $P^{\dagger}=P$ as the corresponding random variables $X_{f}$ and $X_{g}$ would follow the same probability law. As $f(n)\ll\frac{\sqrt{n}}{\log n}$, one would have $f(n)=O_{\varepsilon}(N_{2}(n)^{\frac{1}{2}+\varepsilon})=O_{\varepsilon}(N_{Gal}(n)^{\frac{1}{2}+\varepsilon})$ and thus $\beta_{min}=\beta_{max}=\frac{1}{2}$ which under the ZVG conjecture is equivalent to RH. Maybe the Riemann-von Mangoldt formula would allow to express $f(n)$ and $g(n)$ in terms of the non trivial zeros of the Riemann zeta function.
