Dubner's conjecture

Dubner's conjecture is an as yet (2018) unsolved conjecture by American mathematician Harvey Dubner. It states that every even number greater than 4208 is the sum of two t-primes, where a t-prime is a prime which has a twin. The conjecture is computer-verified for numbers up to $4\times 10^{11}$ .

Even numbers that make an exception are: 2, 4, 94, 96, 98, 400, 402, 404, 514, 516, 518, 784, 786, 788, 904, 906, 908, 1114, 1116, 1118, 1144, 1146, 1148, 1264, 1266, 1268, 1354, 1356, 1358, 3244, 3246, 3248, 4204, 4206, 4208.

The conjecture, if proved, will prove both the Goldbach's conjecture (because it has already been verified that all the even numbers 2n, such that 2 < 2n ≤ 4208, are the sum of two primes) and the twin prime conjecture (there exists an infinite number of t-primes, and thus an infinite number of twin prime pairs).

Whilst already itself a generalization of both these conjectures, the original conjecture of Dubner may be generalized even further:

For each natural number k > 0, every sufficiently large even number n(k) is the sum of two d(2k)-primes, where a d(2k)-prime is a prime p which has a prime q such that d(p,q) = |q − p| = 2k and p, q successive primes. The conjecture implies the Goldbach's conjecture (for all the even numbers greater than a large value ℓ(k)) for each k, and the de Polignac's conjecture if we consider all the cases k. The original Dubner's conjecture is the case for k = 1.
The same idea, but p and q are not necessarily consecutive in the definition of a d(2k)-prime. Again, the Dubner's conjecture is a case for k = 1. It implies the Goldbach's conjecture and the (if we consider all the cases k) are concerned.

Dubner's conjecture

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