New Primality Criteria and Factorizations of 2 m ± 1

Primality tests for 2^{𝑝}±2^{(𝑝+1)/2}+1 using elliptic curves

“…For example, one could use Corollary 1 or Theorem 5 of [1]. These tests apply because p − 1 is divisible by a power of 2 near √ p. These tests determine the primality of p of these three special forms in polynomial time.…”

A Computational Approach to Pocklington Certificates in Type Theory

“…Condition 4 indicates that in order to generate a certificate one needs to be able to partially factorize n − 1 at least till its square root. For our experiment we are using an improved version proposed by Brillhart, Lehmer and Selfridge in [4]. With this new version, we only need to partially factorize till the cube root.…”

“…The existence of a certificate is not direct from Theorem 2 but derives from the exact theorem given in [4] which is a stronger statement: If conditions 5-8 hold then n is prime iff condition 9 holds.…”

Galois theory and primality testing

“…If t = l or 2 then the above test is essentially contained in [3], except for the language that is used. Larger values for t, all dividing 12, were employed by We refer to [5,Section 4] for methods to choose s and t. The number t will usually be somewhat larger than in (6.6).…”