A representation of non-zero elements in finite field (Q1180018)

One of Golomb's conjectures asserts that every non-zero element in the finite field \(GF(q)\) can be written as a sum of two primitive elements. It was proved by some authors that there exists a positive integer \(q_ 0\) such that the conjecture is true if \(q\geq q_ 0\). This paper proves that \(q_ 0\) can be taken as \(6.62\times 10^ 7\) (except \(q=300690391)\), it improves the known results about \(q_ 0\). It is also proved that the conjecture is true if \(q=p^ n\) \((p\) is a prime), \(n\geq 2\), \(q\neq 4\) or \(q=p^ n<10500\), \(p\neq 2,3,5,7,11,13,19,31,43,61\). A computer search is used in the proof.