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

Step 7 of 8: Closing the argument: Bombieri–Vinogradov gives

\liminf(p_{n+1}-p_n)\le 600

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In plain words

Now all the pieces snap together. The classical Bombieri–Vinogradov theorem, known since the 1960s with no need for Zhang's new estimates, guarantees a level of distribution $\theta$ arbitrarily close to $1/2$ . Feeding $\theta=1/2$ and $k=105$ into the threshold from Step 5, together with the numerical fact $M_{105}>4$ from Step 6, gives $r_{105}\ge 2$ : at least two primes among $105$ shifted values, infinitely often.

All that remains is a purely combinatorial fact — that some admissible $105$ -tuple exists packed into an interval of length only $600$ — and the two primes it guarantees are then automatically within $600$ of each other.

\liminf_{n\to\infty}(p_{n+1}-p_n)\le 600

Detailed analysis

By the Bombieri–Vinogradov theorem, the primes have level of distribution $\theta$ for every $\theta<1/2$ (a classical result requiring no input from Zhang's work). Taking $k=105$ , Proposition 4.3 gives $M_{105}>4$ , so Maynard checks directly that with $\theta$ close enough to $1/2$ , $\theta M_{105}/2 > 1$ , so $r_{105}=\lceil \theta M_{105}/2\rceil \ge 2$ (Maynard 2013, §4, proof of Theorem 1.3). By Proposition 4.2 (Step 5), infinitely many $n$ then have at least $2$ of $n+h_1,\dots,n+h_{105}$ simultaneously prime, for any admissible $\mathcal{H}=\{h_1,\dots,h_{105}\}$ .

It remains to exhibit one admissible $105$ -element set of diameter at most $600$ ; such sets can be found by a direct, finite search (choosing $105$ residues avoiding one class mod each small prime), and Maynard confirms one exists. Two of its shifted values being simultaneously prime, infinitely often, means two primes $p_{n+h_i}, p_{n+h_j}$ with $|h_i-h_j|\le 600$ occur infinitely often among consecutive or near-consecutive primes, giving $\liminf_n(p_{n+1}-p_n)\le 600$ (Maynard 2013, Theorem 1.3).

Crucially, this argument uses no estimate beyond the classical Bombieri–Vinogradov theorem: it improves on Zhang's original $70{,}000{,}000$ using purely older technology, made possible entirely by the multidimensional sieve of Steps 3–6.

Terms in this step

Bombieri–Vinogradov theorem: A classical theorem (1965) stating that primes up to $N$ are, on average over moduli $q\le N^\theta$ for any $\theta<1/2$ , distributed among residue classes almost exactly as evenly as the prime number theorem predicts — playing the role that the Generalized Riemann Hypothesis would play if it were known.