REPRESENTATIONS OF ELEMENT AS SUM OF PRIMITIVE ROOT AND LEHMER NUMBER IN Zp.

Let p be an odd prime and Z_p the residue class ring modulo p. In this paper, we study representations of any element of Z_p as the sum of a Lehmer number and a primitive root in Z_p, and give an explicit inequality better than asymptotic formula for the number of representations. From this inequality, we obtained that each element of Z_p can be represented as the sum of a Lehmer number and a primitive root for p > 2.5×10¹⁴. Moreover, using the algorithm we provided, we examined all the cases when p < 10⁶ by computer. We also analyzed the time complexity of the algorithm and illustrated that it is extremely difficult to verify all the cases up to the bound 2.5×10¹⁴, and conjectured that any given element n ∈ Z_p can be represented as the sum of a Lehmer number and a primitive root in Z_p for all primes p except 2, 3, 5, 7, 11, 19, 31. [ABSTRACT FROM AUTHOR]