A prime gap is the difference between two consecutive prime numbers. Formally, if p_n denotes the nth prime number, the nth prime gap is defined as g_n = p_(n+1) - p_n. For example, the gap between 3 and 5 is 2, the gap between 7 and 11 is 4, and the gap between 89 and 97 is 8. Prime gaps are a fundamental object of study in analytic number theory, as their distribution reveals structural patterns in how primes become sparser among larger integers.
Distribution and Growth
Prime gaps tend to increase as numbers grow larger, though not uniformly. The average gap near a large number n is approximately ln(n), as predicted by the prime number theorem. However, the actual gaps fluctuate considerably—sometimes occurring much smaller or larger than average. The largest known prime gaps have been found computationally, with researchers identifying gaps exceeding 1,000 between consecutive primes in the range of very large numbers.
Open Questions
Several fundamental questions about prime gaps remain unsolved. The Cramér conjecture proposes that prime gaps grow no faster than (ln p)² for any prime p, but this remains unproven. The Twin Prime Conjecture, which asserts that infinitely many primes differ by exactly 2, is also unresolved. These questions continue to motivate research in number theory and computational mathematics.