Dubner's conjecture
Dubner's conjecture is an unsolved conjecture by American mathematician Harvey Dubner. It states that every even number greater than 4208 is the sum of two twin primes.[1][2] The conjecture is computer-verified for numbers up to 2 × 1010.[2](p2)
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.[3] (sequence A007534 in the OEIS)
The conjecture, if true, 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 twin primes, and thus an infinite number of twin prime pairs).[2](pp3)
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 Polignac's conjecture if we consider all the cases k. The original Dubner's conjecture is the case for k = 1.[original research?]
- 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 generalized de Polignac's conjecture (if we consider all the cases k) are concerned.[original research?]
References[edit]
- ↑ "Prime Curios! 4208". primes.utm.edu. Retrieved 2023-01-04.
- ↑ 2.0 2.1 2.2 Dubner, Harvey (2000). "TWIN PRIME CONJECTURES". Journal of Recreational Mathematics. 30 (3): 199–205.
- ↑ Sloane, N. J. A. (ed.). "Sequence A007534". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Further reading[edit]
- Jean-Paul Delahaye (June 2002), Nombres premiers inévitables et pyramidaux, Pour la Science, issue 296, pp. 98–102
Template:Prime number conjectures
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