Proven approximations: bounded gaps and Chen's theorem
The argument
Two families of theorems establish weakenings of the conjecture in each of its two directions. Relaxing the gap, infinitely many prime pairs differ by at most a fixed constant, and specifically infinitely many prime pairs differ by at most 246; relaxing primality instead, infinitely many primes p have p + 2 prime or a product of two primes. Given that the conjecture is the common sharpening of both (gap exactly 2, both members prime), these results show that the behaviour it asserts is real up to a bounded gap or one extra prime factor, which raises confidence in the conjecture without proving it.
Does it work?
Not yet evaluated.
Premises
- UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →There are infinitely many pairs of primes that differ by at most 246.
- UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →There are infinitely many pairs of primes whose difference is at most a fixed constant.
- UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →There are infinitely many primes p such that p + 2 is either prime or a product of two primes.
Where it is made
Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.
No source is recorded as making it.
Earlier ratings
- Sep 16, 2026
The premises are established theorems and are not in doubt: Chen's result that infinitely many primes p have p + 2 prime or a product of two primes and the bounded-gaps theorems that infinitely many prime pairs differ by at most a fixed constant, indeed by at most 246. What they yield is corroboration rather than implication: neither a bounded gap nor an almost-prime companion entails a gap of exactly 2 with both members prime, and the parity obstruction in sieve theory is precisely why these methods stop short of the conjecture. The argument therefore raises confidence in the conjecture by showing its asserted pattern survives every relaxation that has been proved, and it can do no more than that.