Proof of Golomb’s conjecture in Fq with Γ$-pseudorandom sequences

This article offers a short proof of Golomb’s conjecture, and then our results show that the sequences are pseudorandom in F2. 2020 Mathematics Subject Classifications: 11A41, 11A07, 12E20, 11B50


Introduction
Question 10208b (1992) of the American Mathematical Monthly asked: does there exist an increasing sequence {a k } of positive integers and a constant B > 0 having the property that {a k + n} contains no more than B primes for every integer n? If it turns out that a positive answer to this question became known as Golomb's conjecture [1].
Let F q denote the finite field of order q, where q = p n , and p is a prime, if n=1, Golomb's conjecture equivalent to: Conjecture 1. Let α and β be two primitive roots of f (x)(mod p) in F q , then Moreno and Sotero [2] proved that Golombs conjecture is true for all q < 2 60 . Golomb [3] pointed out that the conjecture associated with sequences and prime numbers. Elsholtz [4] proved in 2017 that Golomb's conjecture was false. Elsholtz gave an example to explain the Fermat numbers contains at least B + 1 primes, but his research has shown that we cannot conclude that there is any fixed n such that the sequence {2 2 i +n} contains innitely many primes. Certainly! the conjecture is true for the special case.
Because of pseudorandom sequences unique characteristic they have been widely used in many fields. In this paper we prove that the Golombs conjecture in F q , and we show that a new pseudorandom sequence (denote Γ ) in F 2 is existent by Golomb conjecture and additive groups.
is prime, then is prime.
Proof. If = 1, then p is not prime. Let be composite and let m be divisor. Since = km satisfies 1 < k < , and that We have and so p is not prime, a contradiction.
For our next discussion, Γ numbers are defined by As an application of the Γ numbers, we give the following theorem. Theorem 2. 3 is a primitive root of Γ 7 (or Γ 11 ).

(8)
The same result is primitive root of modulus 34961. Hence as desired.
Combining the Theorem 2.3 and Theorem 2.4, we obtain the following corollary.
Corollary 1. If Γ is prime and is primitive root of f(x)(mod Γ ), then 3. Γ -pseudorandom sequences Definition 1. Let is prime and let µ is primitive root of modulus Γ such that Eq. (5). We say that Γ is a period of pseudorandom sequence if X = [a 0 a 1 a 2 · · · a p−2 a p−1 ] (a i ∈ F q ) (12) with the following properties: for where 1 < i < p−1.
Definition 2. Let η be an additive group of F 2 (for +1 and −1). Then multiplicative group is isomorphic which constitutes by these two integers, and we have Our new result is the following theorem. Theorem 5. Suppose that the pseudorandom periodic sequence Γ acts transitively on the F 2 . Then (17) Proof. By Definition 1. First, we have Let j ≡ 0 (mod Γ ). We define X by Next let f (y) be a minimal polynomial of X with X∈ G(f ). Now If f (x) is nth order primitive polynomial f (y) = c n y n + c n−1 y n−1 + · · · + c 1 y + c 0 (c 0 c n = 0). (20) Then X satisfies the homogeneous linear difference equation of nth order We have To find that the linear recursive relation for Eq.(21) so that X moves that S steps to the left, we have and If since period X is Γ , then But nonzero sequence is Γ in G(f ), so 1 there are n=1 (−1) n+1 2 −n−1 times in a period in X + T s (X) and 0 there are n=1 (−1) n+1 2 −n−1 − 1 times in a period in X + T s (X).
Finally, by Definition 2, we have as desired.