Polignac's conjecture

About 177 years old · traced to 1849

Let $p_1 < p_2 < p_3 < \cdots$ denote the increasing sequence of all prime numbers, so that $p_k$ and $p_{k+1}$ are consecutive primes for every $k \ge 1$ . For an even positive integer $n$ and a real number $x \ge 2$ , put

\pi_n(x) = \#\{\, k \ge 1 \;:\; p_k \le x, \; p_{k+1} - p_k = n \,\}.

For every even positive integer $n$ there are infinitely many indices $k$ with $p_{k+1} - p_k = n$ ; equivalently, $\pi_n(x) \to \infty$ as $x \to \infty$ for every even $n \ge 2$ . That is, for every even $n \ge 2$ there are infinitely many primes $p$ such that $p + n$ is \prime and no \prime lies strictly between $p$ and $p + n$ .

References

Primary source

Wikipedia

Additional references

Wikipedia, Polignac's conjecture, the article this problem comes from.

Posted 2 months ago

Progress summary

Refreshed 1 month ago

Claimed progress

The conjecture remains unproved, despite major progress on bounded prime gaps and unsupported claims of a complete proof.

Proposed by Alphonse de Polignac in 1849, the conjecture predicts infinitely many consecutive prime pairs at every prescribed positive even gap. No particular even gap has been proved to occur infinitely often as a consecutive-prime gap.

Known results

Zhang, 2013: some even gap $n<70{,}000{,}000$ occurs infinitely often.
Maynard and Tao, 2013--2014: bounded-gap methods reduced the unconditional bound to $246$ .
Chen, 1966: for every even $h$ , infinitely many primes $p$ have $p+h$ prime or semiprime.
Sawin and Shusterman, 2019: the analogous statement holds for prime polynomials over finite fields, not ordinary primes.

August 12 claimed proof

A post dated August 12 claimed “Proof of Polignac’s conjecture! QED.” No proof, verification, or independent confirmation is supplied in the scan. Separately, Sankei et al. published a claimed logical proof on March 16, 2024, while the article itself acknowledged that the conjecture remained unproven.

Current status (as of August 2026): Bounded gaps are known, but Polignac’s conjecture for every even $n$ remains open; the August 12 proof claim is unverified.

Sources

No solutions have been posted yet.