Twin-Prime Conjectures

The conjecture, commonly attributed to Alphonse de Polignac in 1849, asserts that infinitely many primes $p$ have $p+2$ prime.

Known results

• Zhang (2013): infinitely many prime pairs have a fixed gap below $70{,}000{,}000$.
• Maynard and Tao (2013), followed by Polymath (2014): the unconditional bound was reduced to $246$, not $2$.
• Chen (1973): infinitely many primes $p$ have $p+2$ with at most two prime factors.
• Arenstorf’s purported proof was retracted; Czelakowski’s proof was retracted in 2022 after an irreparable error was identified.

August 2026 conditional reduction and claimed proofs

On August 23, 2026, Dacomb Bierton’s deposit Structural Reduction of Two Twin-Prime Conjectures and Conditional Resolution under Hypothesis H reported a conditional reduction, not an unconditional solution. Preprints from 2017 and 2019 claim unconditional proofs, but no independent verification or accepted resolution is recorded.

Current status (as of August 2026): the twin-prime conjecture remains open; bounded gaps and conditional implications are established, while the reported unconditional proofs remain unverified.