Lower bounds for maximal consecutive prime gaps
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How large must the largest consecutive prime gap below X be for every sufficiently large X? The target lower order is log X·(log₂X)²·log₄X/(log₃X)², where log_j denotes j-fold iterated logarithm.
Why it matters and what counts as progress
Why it matters: It refines the asymptotic scale guaranteed for extreme prime gaps, beyond merely showing gaps are unbounded.
What counts as progress: A stronger asymptotic lower bound for the largest prime gap below every sufficiently large X, with all iterated logarithm factors controlled.
How the problem developed
- 1938 · Lower bound [1]
Robert A. Rankin Improved the classical construction of long gaps between consecutive primes.
- 2018 · Lower bound [2]
Kevin Ford, Ben Green, Sergei Konyagin, James Maynard and Terence Tao Improved the asymptotic lower bound for the largest consecutive-prime gap below X.
- 25 Aug 2026 · Lower bound claim [3]
DottedCalculator, GPT-5.6 Sol Posted a residue-class construction claiming a stronger lower bound for the maximal gap G(X).
- 3 Sept 2026 · Lower bound claim [4]
OpenAI Posted a further claimed lower bound for G(X), with a different iterated-logarithm factor.
AI contributions 2
Improved Long Gaps Between Primes
For all sufficiently large X, the largest consecutive prime gap below X is at least a constant times log X (log₂ X)² log₄ X/(log₃ X)². The argument uses a short-translates proposition and a weighted sieve.
A Tilted Residue-Class Construction for Long Prime-Free Intervals
The tilted residue-class sieve gives Y(X)≫X log X/log₃ X for covering and maximal prime gaps G(T)≫log T log₂ T/log₄ T for all sufficiently large T.
Sources
- Robert A. Rankin, The difference between consecutive prime numbers
- Kevin Ford, Ben Green, Sergei Konyagin, James Maynard and Terence Tao, Long gaps between primes
- GPT 5.6 Sol, tilted residue-class construction
- Long gaps between primes paper
- Formalization status and theorem alignment
- Indexed-prime formal statement
- Alexeev, formalization of the tilted covering and gap bounds
- Erdős Problems, tilted-sieve claim and discussion
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