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I would like to propose the following conjecture

The PKD Conjecture (PKD)

Let $p,d$ be positive integers with $\gcd(p,d)=1$. There exists a function $$ f:\mathbb{N}\to\mathbb{N}, \quad f(N)<N, $$ such that there always exists an integer $k$ satisfying $$ 0 < k \le f(\max\{p,d\}), \quad q = pk + d \text{ is prime}. $$

Special cases

  • Case $d<p$: This reduces to the existence of the smallest prime in the residue class $d \pmod p$. This is related to Linnik’s theorem on the least prime in an arithmetic progression.

  • Case $d>p$: Then the conjecture asserts that a prime appears in the interval $$ (d,\, d+p\cdot f(\max\{p,d\})). $$

  • Case $p=1$: This is equivalent to Cramér’s prediction, i.e. that there is a prime in the interval $$ (d,\, d+f(d)). $$

Heuristic

A reasonable growth for $f$ is expected to be $$ f(N) \sim 5.7 (\log N)^2, $$ which is consistent with Cramér’s conjecture on prime gaps.

Motivation

This conjecture can be viewed as a “meta-conjecture” that generalizes both:

  • the distribution of primes in arithmetic progressions (Linnik-type problems), and
  • the distribution of primes in short intervals (Cramér-type problems).

It suggests a unified framework for understanding prime distribution both along progressions and within bounded gaps.

Question

  • Has this conjecture (or similar formulations) appeared in the literature?
  • Are there known partial results that could support or contradict it?

Any references or insights would be greatly appreciated. link:https://doi.org/10.5281/zenodo.17327717

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    $\begingroup$ This is AI junk. $\endgroup$ Commented Nov 8 at 13:38

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For large $m:=\max\{p,d\}$, the "PKD conjecture" follows from a conjecture of Hugh Montgomery recorded as (17.5) in Iwaniec-Kowalski: Analytic number theory: for any $\varepsilon>0$, $$\psi(x;p,d)=\frac{x}{\varphi(p)}+O_\varepsilon(p^{-1/2}x^{1/2+\varepsilon}).$$ It is straightforward to show that this is equivalent to: for any $\varepsilon>0$, $$\theta(x;p,d)=\frac{x}{\varphi(p)}+O_\varepsilon(p^{-1/2}x^{1/2+\varepsilon}).\label{1}\tag{$\ast$}$$ Let us assume this, and let us choose (as we can) $f(N):=N-1$ in the "PKD conjecture". Then the desired conclusion is the existence of $0<k\leq m-1$ such that $pk+d$ is prime. In other words, $$\theta(pm-p+d;p,d)-\theta(d;p,d)>0.$$ However, \eqref{1} implies that $$\theta(pm-p+d;p,d)-\theta(d;p,d)=\frac{pm-p}{\varphi(p)}+O_\varepsilon(p^\varepsilon m^{1/2+\varepsilon}).$$ On the right-hand side, the main term is at least $m-1$, while the error term is $o(m)$ for any $\varepsilon<1/4$, whence the expression is positive for large $m$.

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    $\begingroup$ Thank you very much for your detailed answer. It seems that my conjecture is not contradictory $\endgroup$ Commented Nov 9 at 3:02

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