In the set of prime numbers, the equation p-q = 2 has multiple solutions. There is a hypothesis that this equation has infinitely many solutions. What happens if 2 is replaced by 2k?
Ths proposed Polignac's conjecure (1849) is clearly, a paricular case of Dickson's conjecture stated by Dickson (Dickson, L. E. (1904), "A new extension of Dirichlet's theorem on prime numbers", Messenger of mathematics, vol. 33, pp. 155–161) which asserts that for a finite set of linear forms a1 + b1n, a2 + b2n, ..., ak + bkn (ai and bi are inegers for all i = 1, 2,..., k and bi ≥ 1), there are infinitely many positive integers n for which they are all prime numbers. The proposed Polignac's conjecure (1849) is clearly, a paricular case of Dickson's conjecture stated by Dickson (Dickson, L. E. (1904), "A new extension of Dirichlet's theorem on prime numbers", Messenger of mathematics, vol. 33, pp. 155–161) which asserts that for a finite set of linear forms a1 + b1n, a2 + b2n, ..., ak + bkn (ai and bi are inegers for all i = 1, 2,..., k and bi ≥ 1), there are infinitely many positive integers n for which they are all prime numbers.
It was noticed in https://en.wikipedia.org/wiki/Polignac%27s_conjecture that, assuming the Elliot-Hailberstam conjecture and its generalized form, the Polymath project wiki (https://en.wikipedia.org/wiki/Polymath_Project) states that he values 2k has been reduced to 12 and 6, respectively.[6]
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