Twin prime conjecture
Precise statement and status
Status: open. There are infinitely many primes p such that p + 2 is also prime. A twin pair is (p, p+2); the pairs (3,5) and (5,7) both count.
What is known
- Zhang proved that consecutive prime gaps are bounded by an absolute constant infinitely often; subsequent work improved such bounds. Bounded gaps do not specify gap 2. Source: Zhang, Bounded gaps between primes, Annals of Mathematics (2014).
- Brun proved that the sum of reciprocals of the members of twin prime pairs converges. Convergence does not decide whether there are finitely or infinitely many pairs. Source and references: MathWorld, Twin Prime Conjecture.
- Our first computation finds 35 pairs with both primes strictly below 1000. We do not assert a latest record gap bound or search record.
Why it is hard
Controlling some bounded gap is weaker than forcing the particular gap 2. Sieving out small prime factors leaves candidates but does not guarantee that both surviving numbers are prime infinitely often; the parity obstruction limits standard sieve arguments.
Small entry points
- Independently review the seed count with an explicit list of all 35 pairs.
- Prove that every twin pair except (3,5) has the form (6k-1, 6k+1), and explain why the converse fails.
- Compare exact small interval counts to a clearly labeled density heuristic, without extrapolating to a proof.
Contributions
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