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Bounded Prime Gaps

Bounded prime gaps refer to the property that there exists a finite number $H$ such that infinitely many pairs of consecutive primes differ by less than $H$. This concept is central to progress on the twin-prime-conjecture, which posits that $H=2$ (i.e., there are infinitely many twin primes).

Key Developments

• The existence of bounded gaps between primes was proven in 2013, marking a major breakthrough in analytic number theory.
• Prior to this, it was unknown whether the gap between consecutive primes remained bounded as primes approached infinity.
• The result does not resolve the twin-prime-conjecture but establishes that prime clusters are denser than previously proven bounds allowed.

Yitang Zhang’s Breakthrough

• In 2013, Yitang Zhang proved that there exists a constant $H\le 70,000,000$ such that infinitely many pairs of primes differ by no more than $H$.
• This was the first time a finite bound was established for prime gaps, shifting the problem from existence of any bounded gap to optimizing the value of $H$.
• Subsequent collaborations (Polymath project) and refinements by James Maynard significantly reduced this bound.

Related Concepts

• prime-number-theorem: Describes the asymptotic distribution of primes; implies average gaps grow logarithmically, but does not preclude bounded small gaps occurring infinitely often.
• Goldbach’s Conjecture: Another unsolved problem concerning additive properties of primes.
• Sieve Methods: Technical tools (e.g., Selberg sieve) used in Zhang’s proof to isolate prime pairs.

Sources & References

• Yitang Zhang’s Proof: Bounded Prime Gaps and the Twin Prime Conjecture — Veritasium documentary on Zhang’s work and its context.
• Yitang Zhang’s Proof: Bounded Prime Gaps and the Twin Prime Conjecture — Detailed summary of the historical and technical narrative.