Fast Algorithms on Primality Testing for Numbers 255 ⋅ 2^n ± 1

In this paper, we study the problem of primality testing for numbers of the form h · 2ⁿ ± 1, where h <; 2ⁿ is odd, and n is a positive integer. We describe explicit Lucasian primality tests for these numbers in certain cases, which run in deterministic quasi-quadratic time. In particular, we implement the explicit algorithms related to numbers M_n = h · 2ⁿ - 1 for the case h =255. And we find large prime numbers which are more than 512 bits efficiently. The algorithms of Bosma (1993), Berrizbeitia and Berry (2004), Deng and Huang (2016) can not generate these prime numbers, even can not test the primality of these numbers.