Prime Gaps and Recent Progress

Introduction to Prime Gaps

Prime gaps: The difference between one prime and the next.
Notation: Given a prime (P_n), the prime gap is the number of steps to the next prime.
First prime gap: 1, then all subsequent prime gaps are even numbers since all primes after 2 are odd.

Twin Prime Conjecture: There are infinitely many pairs of primes that are distance 2 apart.
- The average prime gap increases roughly like the logarithm of the primes.
- The prime gap might occasionally be as small as 2.
Breakthrough by Yitang Zhang (2013): Demonstrated there are infinitely many pairs of adjacent primes with a gap bounded by a large fixed constant.
- Initial bound: 70 million.
- Later improved via collaborative online project (Polymath Project) to 4,000 and then to 246.

Finding large prime gaps = Identifying long sequences of consecutive composite numbers.
Simple Construction: Using (n!) to generate a sequence of composite numbers.
More Efficient Construction: Using prime factorial (primorial) provides better bounds.

Prime Number Theorem (1896): Number of primes up to a large number (x) is approximately (x / \log(x)).
Unconditional Bound: Prime gaps are at most a constant times (P_n^{0.55}).
Conditional Bound (Assuming Riemann Hypothesis): Improves to (P_n^{0.5} \log(P_n)).
Cramér's Conjecture: Prime gaps can be as large as (\log^2 P_n) infinitely often.

First non-trivial results in 1931: Managed to show prime gaps larger than the trivial logarithmic bound.
Progress stalled for 70 years until recent breakthroughs.
Ford-Green-Konyagin-Tao (2014): Proved can take (C) arbitrarily large (improving an old result by Erdős offering $5,000 for a significant advancement).
Maynard (2014): Found a method to prove the same result independently.

Pigeonhole Principle: If the total probability of events exceeds one, at least one event must happen twice.
Weighted Sieve Method: Choose a clever set of weights to optimize probabilities.
Selberg Sieve: Weights determined by considering multiples and factors, squared to ensure non-negativity.
Bombieri-Vinogradov Theorem: Helps count primes in arithmetic progressions.

Yitang Zhang: Used advanced theorems about primes in arithmetic progressions, extending previous bounds.
James Maynard: Introduced refined sieve methods, simplifying previous approaches.
Polymath Project: Collaborated effort improving results using computational and theoretical advancements.

Same basic methods used for small and large prime gap results now being applied to other number theory problems.