Worked solution: Maynard's multidimensional sieve: bounded gaps between primes (2013)

Step 8 of 8: Beyond pairs: arbitrarily many primes, and the road to 246

In plain words

The same machinery that gave $2$ primes among $105$ shifts also works, with a larger $k$ , for $3$ , $4$ , or any number $m$ of primes at once: since $M_k$ grows at least like $\log k$ , choosing $k$ large enough always forces $m$ primes into a bounded window, for every $m$ . This is the general Theorem 1.1 of Maynard's paper — arbitrarily many primes, clustered together, infinitely often.

The specific constant $600$ was never claimed to be optimal — Maynard says so explicitly — and, indeed, a large collaborative effort called Polymath8b pushed the same method's numerics further within months, bringing the bound down to $246$ , still the best known unconditional bound today. The full twin prime conjecture (bound $2$ , not just some bounded number) remains open.

\liminf_{n\to\infty}(p_{n+m}-p_n) \ll m^3 e^{4m}

Detailed analysis

Because $M_k>\log k-2\log\log k-2$ for large $k$ (Step 6), and $\theta$ can be taken close to $1/2$ unconditionally, the threshold $r_k=\lceil\theta M_k/2\rceil$ grows like $\tfrac14\log k$ as $k\to\infty$ . Choosing $k$ appropriately in terms of a target $m$ , Maynard derives Theorem 1.1: for every $m\in\mathbb{N}$ , $\liminf_n(p_{n+m}-p_n)\ll m^3 e^{4m}$ (Maynard 2013, Theorem 1.1) — arbitrarily many primes in a bounded window, for every $m$ , unconditionally.

Maynard is explicit that neither $600$ (Theorem 1.3) nor the $m^3e^{4m}$ growth rate (Theorem 1.1) is claimed optimal: 'by performing further numerical calculations our method could produce a better bound' (Maynard 2013, §1). This is exactly what happened: the Polymath8b project (2014), a large online collaboration combining Maynard's and Terence Tao's independently discovered versions of the same method with extensive computer search over admissible tuples, reduced the bound to $\liminf_n(p_{n+1}-p_n)\le 246$ , the best unconditional bound known today.

The twin prime conjecture itself — that the bound can be taken all the way down to $2$ — remains completely open; Maynard's method, even combined with the Elliott–Halberstam conjecture (an unproven strengthening of Bombieri–Vinogradov to $\theta$ close to $1$ ), gets only as far as $\liminf_n(p_{n+1}-p_n)\le 12$ (Maynard 2013, Theorem 1.4). Closing this remaining gap to $2$ would need fundamentally new ideas beyond sieve theory as currently understood.

Terms in this step

Elliott–Halberstam conjecture: An unproven strengthening of the Bombieri–Vinogradov theorem, conjecturing that primes have level of distribution $\theta$ for every $\theta<1$ , not just $\theta<1/2$ . It is a natural target for sieve methods but remains open.

Knowledge used in this step

Bounded gaps between primes (Zhang, Maynard)