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Prime Gaps

Prime gaps refer to the difference between consecutive prime numbers. While the Prime Number Theorem describes the asymptotic distribution of primes, the behavior of these gaps reveals significant irregularities in their density.

Key Properties

• Gaps can be arbitrarily large. For any integer $n$, there exists a sequence of $n$ consecutive composite numbers.
• The Twin Prime Conjecture posits that gaps of size 2 occur infinitely often, though this remains unproven.
• Prime Gap statistics are critical in understanding the distribution of prime numbers and have implications for cryptography and computational-complexity.

Constructive Proof of Arbitrarily Large Gaps

A standard method to demonstrate that prime gaps can exceed any given bound involves the use of factorials.

• Method: Consider the sequence of integers $n!+2,n!+3,\dots ,n!+n$.
• Logic: For each $k$ in the range $2\le k\le n$, $k$ divides $n!$. Consequently, $k$ also divides $n!+k$.
• Result: Since each term in the sequence has a divisor other than 1 and itself, all terms are composite. This creates a gap of at least $n-1$ between the prime preceding $n!+2$ and the prime following $n!+n$.
• Implication: As $n\to \infty$, the gap size $\to \infty$, proving that there is no maximum prime gap.

For a detailed breakdown of this constructive approach, see Constructive Proof of Arbitrarily Large Prime Gaps Using Factorials.

References

• Constructive Proof of Arbitrarily Large Prime Gaps Using Factorials (Numberphile, 2026-09-17)