Golomb猜想在有限域GF(p^n)中的若干结果

THE RESULT OF THE GOLOMB CONJECTURE IN SOME FINITE FIELDS

本文用比较简捷的方法获得了Golomb猜想在有限域GF(p^n)中成立的几个结果。这些结果对于不太大的有限域GF(p^n)来说是有意义的,对这些有限域的代数构造是有价值的。

In this paper, we have obtained some results of the Golomb's conjecture, i. e., we have proved 4 theorems as follows. THEOREM 1: If p is an odd prime when φ(p^n-1)/p^n-1>3/8, the Golomb's conjecture is correct in the finite field GF(p^n), where φ(m) is the Euler's function. THEOREM 2: If p is an odd prime, and p^n≡1 (rood 4), when φ(p^n-1)/(p^n-1)>1/4, the Golomb's conjecture is correct in the finite field GF(p^n). From Thereto 1 and Theorem 2, we can obtain the following theorem 3 and theorem 4. THEOREM 3: If an odd prime p satisfies φ(P-1)/(p-1)>3/8, when the odd prime q>1/2p^2 1/(2log(φ(p-1)/(p-1)/(3/8)), (p-1, q)=1, the Golomb's conjecture is correct in the finite field GF(p^q), there are two primitive elements α, and β satisfy α+β=1 in the finite field GF(p^q). THEOREM 4: If an odd prime p≡1 (rood4) satisfies φ(p-1)/(p-1)>1/4, when the odd prime q>1/2p^2 1/(21og(φ(p-1)/(p-1)/(1/4)), (p-1, q)=1, the Golomb's conjecture is correct in the finite field GF(p^q).