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on Linnik's constant ($C_{65}$)

The statement

For integers $q\ge 2$ and $a$ with $\gcd(a,q)=1$ , let $P(a,q)$ denote the least prime in the arithmetic progression $a \bmod q$ . [Xyl2011-def-Paq]

Linnik's theorem asserts that there exist constants $C,L>0$ such that

P(a,q)\ \le\ C\,q^{L} \qquad (\gcd(a,q)=1),

uniformly for all $q\ge 2$ . [Xyl2011-def-Linnik]

We define

C_{65}\ :=\ L_{\mathrm{Lin}},

where $L_{\mathrm{Lin}}$ is the infimum of all exponents $L$ for which such a bound holds (with some constant $C$ independent of $a,q$ ). [Xyl2011-def-Linnik]

The best known exponent reported in a modern source is

L_{\mathrm{Lin}}\ \le\ 5.

[MMT2024-ub-5]

A trivial lower bound is $L_{\mathrm{Lin}}\ge 1$ , since for $a=1$ the first candidate prime is at least $q+1$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.