The positive integer difference g_n = p_{n+1} − p_n between two successive prime numbers p_n and p_{n+1}; a local measure of how far apart consecutive primes lie.

Principle

Prime gaps reflect the distribution of primes: average gaps grow like log p by the prime number theorem, while local behavior is governed by sieving constraints, probabilistic models, and arithmetic structure that permit both small gaps (twin primes) and arbitrarily large gaps.

Demonstration

Examples: between primes 7 and 11 the gap is 4; between 89 and 97 the gap is 8. The average gap near x is approximately log x; e.g., around x≈100, log x≈4.6, consistent with observed local gaps.

Misapplication

Treating prime gaps as monotone increasing with p, or assuming that small average gap implies absence of long gaps; misusing probabilistic models as exact predictions rather than heuristics.

Consequence

Understanding gaps informs conjectures (twin primes, bounded gaps), proofs about prime distribution, and practical estimates for locating primes and cryptographic prime generation; claims about large or bounded gaps have major theoretical implications.

Reversal

Contrast by studying prime clusters or constellations (patterns of small gaps) which emphasize local closeness rather than separation; inversion also considers maximal prime gaps over intervals instead of minimal ones.

Boundary

Defined only for consecutive prime pairs in the sequence of primes (including the gap after 2); excludes nonprime indices and differences between nonconsecutive primes; asymptotic statements rely on sufficiently large primes where the prime number theorem applies.

Semantic Tension

Tension between global average growth (logarithmic) and wild local fluctuations (arbitrarily large gaps versus frequently small gaps), and between probabilistic models predicting typical behavior and deterministic constructions that produce extreme gaps.

Synthesis

A prime gap is the local difference between successive primes that balances an average logarithmic growth with irregular local variation; it is both a measurable statistic for prime distribution and a focal object for deep conjectures and constructive examples.