# Predicting maximal gaps in sets of primes

## TL;DR

This paper develops heuristic formulas to predict the growth of maximal gaps between primes in specific residue classes and prime k-tuples, supported by extensive computational evidence up to 10^{14}.

## Contribution

It introduces new trend formulas for maximal prime gaps in residue classes and prime k-tuples, validated by large-scale computations, and explores their distribution and conjectural properties.

## Key findings

- Simple trend formula for k≥2 matches computational data.
- Refined formula improves predictions for k=1, including all primes.
- Maximal gaps distribution approximates Gumbel extreme value distribution.

## Abstract

Let $q>r\ge1$ be coprime integers. Let ${\mathbb P}_c={\mathbb P}_c(q,r,{\cal H})$ be an increasing sequence of primes $p$ satisfying two conditions: (i) $p\equiv r$ (mod $q$) and (ii) $p$ starts a prime $k$-tuple with a given pattern ${\cal H}$. Let $\pi_c(x)$ be the number of primes in ${\mathbb P}_c$ not exceeding $x$. We heuristically derive formulas predicting the growth trend of the maximal gap $G_c(x)=\max_{p'\le x}(p'-p)$ between successive primes $p,p'\in{\mathbb P}_c$. Extensive computations for primes up to $10^{14}$ show that a simple trend formula $$G_c(x) \sim {x\over\pi_c(x)}\cdot(\log \pi_c(x) + O_k(1))$$ works well for maximal gaps between initial primes of $k$-tuples with $k\ge2$ (e.g., twin primes, prime triplets, etc.) in residue class $r$ (mod $q$). For $k=1$, however, a more sophisticated formula $$G_c(x) \sim {x\over\pi_c(x)}\cdot\big(\log{\pi_c^2(x)\over x}+O(\log q)\big)$$ gives a better prediction of maximal gap sizes. The latter includes the important special case of maximal gaps in the sequence of all primes ($k=1$, $q=2$, $r=1$). The distribution of appropriately rescaled maximal gaps $G_c(x)$ is close to the Gumbel extreme value distribution. Computations suggest that almost all maximal gaps satisfy a generalized strong form of Cramer's conjecture. We also conjecture that the number of maximal gaps between primes in ${\mathbb P}_c$ below $x$ is $O_k(\log x)$.