Deterministic primality test for numbers of the form $A^2.3^n+1$, $n \ge 3$ odd

Abstract

Abstract. We use a result of E. Lehmer in cubic residuacity to find an algorithm to determine primality of numbers of the form A 2 3 n + 1, n odd, A 2 < 4(3 n + 1). The algorithm represents an improvement over the more general algorithm that determines primality of numbers of the form A.3 n ± 1, A/2 < 4.3 n − 1, presented by Berrizbeitia and Berry (1999).

On the Analytical Properties of Prime Numbers

“…In addition to that, the result are expressed a polynomial that highlights the properties of those numbers, and it can also be used as a primitive test to discover those numbers . For a discussion of such issues see [3,8,9,[11][12][13][14] on there are several numbers studied A3 n AE 1, k2 n þ 1, 2 n AE 1 and close to these formulas.…”

“…There are many works that discuss Broth's theorem and numbers. Case p ¼ 3 studied by W. Bosma [4] and A. Guthmann [5] Also, for a discussion on the Broth numbers, see H.C Williams [6,7] P. Berrizbeitia [8,9]. The purpose of this work is to study the numbers in model ma m þ bm þ 1 and p ¼ ba þ 1 where a, b > 1 ∈  and p ¼ aq þ 1, a, q ∈  where q is an odd prime number, In addition, tests for Fermat and Mersenne numbers are presented and the study of the relationship between two prime numbers and a polynomial with finite properties.…”

A primality test for 𝐾𝑝ⁿ+1 numbers

“…In recent times the most active researcher looking for primality criteria for numbers of the form N = Kp n + 1 has been P. Berrizbeitia. Berrizbeitia and his collaborators have found very efficient criteria for this kind of numbers for a variety of primes p (see [5,6,7]). Even though similar criteria had been previously presented by H.C. Williams and his collaborators (see [23,22]), the methodology used by Berrizbeitia et al shows more clear and efficient.…”