Abstract
In this paper, an interesting third-order linear recurrence formula is presented by using elementary and analytic methods. This formula is concerned with the calculating problem of the hybrid power mean of a certain two-term exponential sums and the cubic Gauss sums. As an application of this result, some exact computational formulas for one kind hybrid power mean of trigonometric sums are obtained.
Highlights
The reason why we focus on the calculation of (7) is that the problem is closely related to the number of the solutions of some congruence equation. ese contents play a very important role in study of some famous analytic number theory problems, such as Waring problem and Goldbach conjecture
Rough the study, it is found that the problem we studied is closely related to integer 3
If p is a prime with p ≡ 1 mod 3, for any third-order character ψ mod p, we have the identity p− 1
Summary
If 3 is a cubic residue mod p, for any integer k ≥ 3, one has the thirdorder linear recurrence formula: Hk(1, p) 3pHk− 2(1, p) + dpHk− 3(1, p),. If 3 is a cubic residue modulo p, there exists a beautiful third-order linear recurrence formula for Uk(p), and the first three terms U0(p), U1(p), and U2(p) are integers. If 3 is a cubic residue modulo p, for any integer k ≥ 3, we have the third-order linear recurrence formula. For any fixed positive integer h ≥ 5, whether there is a third-order linear recurrence formula for the hybrid power mean.
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