On the quartic Gauss sums and their recurrence property

The main purpose of this paper is, using the method of trigonometric sums and the properties of Gauss sums, to study the computational problem of one kind of congruence equation modulo an odd prime and give some interesting fourth-order linear recurrence formulas.


Introduction
Let q ≥  be a positive integer. For any positive integer k and integer m, the kth Gauss sums G(m, k; q) are defined as where p is an odd prime, χ denotes a Dirichlet character mod p and k denotes the Big O notation dependent on k.
We also mention that the fourth power mean value of G(m, k; q) was well explored by Zhang and Liu [], in which some sharp asymptotic formulas can be found.
On the other hand, Yang and Tang [] studied a number of solutions of the congruence equation x  + y  ≡ c mod n with (xy, n) =  and gave an exact computational formula for it.
Let s be a positive integer and p be an odd prime with p ≡  mod . Let M s denote a number of solutions of the equation where U  = , U  = p - and U  = (p -)d with d being uniquely determined by p = d  + b  and d ≡  mod . Some related results can also be found in [-]. Now we consider a similar problem: Let n be a positive integer and p be an odd prime with p ≡  mod . Let M n (p) denote a number of solutions of the congruence equation It is natural to ask whether there exists an exact computational formula for M n (p) when n is a positive integer and p is an odd prime?
As far as we know, it seems that no one has studied this problem yet, at least we have not seen any related result before. The problem is interesting because it can help us to understand more accurate information of the quartic Gauss sums.
In this paper, we shall use the method of trigonometric sums and the properties of Gauss sums to study this problem and give some interesting computational formulas. For the sake of convenience, first we let B(p) = p- a= ( a+a p ), a denotes the solution of the equation ax ≡  mod p, and ( * p ) denotes the Legendre symbol mod p. Then we have the following theorem.
Theorem  Let p = k +  be a prime, U n (p) = M n (p)p n- . Then, for any positive integer n ≥ , we have the fourth-order linear recurrence formula Theorem  Let p = k +  be a prime, U n (p) = M n (p)p n- . Then, for any positive integer n ≥ , we have the fourth-order linear recurrence formula where the first four terms are U  (p) = , U  (p) = (p -), U  (p) = (p -)B(p) and U  (p) = p(p -) + (p -)B  (p).
From these theorems we may immediately deduce the following two corollaries. Note the estimate for character sums from Corollary  and Corollary  we can also deduce the following corollary.
Corollary  Let p be an odd prime with p ≡  mod . Then we have the asymptotic formulas Remark Let p be an odd prime with p ≡  mod , it is clear that we have the identity (see where r is a quadratic non-residue mod p, that is to say, ( r p ) = -. The above identity implies that |B(p)| is a constant depending only on p and |B(p)| ≤  √ p.
For prime p = k +  and positive b with  ≤ b ≤ p -, we have the identity It is very easy to prove that M n- (p) = p n- and M n (p) = p n- + (-) n p n- (p -).

Several lemmas
In this section, we give some lemmas which are necessary in the proofs of our theorems. Hereinafter, we shall use some properties of the classical Gauss sums, all of them can be found in reference [], so they will not be repeated here. First we have the following lemma.
Lemma  Let p be an odd prime with p ≡  mod , ( * p ) = χ  denotes the Legendre symbol mod p. Then, for any integer b with (b, p) = , we have the identities where R(p) = -, if p = k + ; and R(p) = , if p = k + .
Proof From the definition of the quadratic residue mod p and the properties of the Legendre symbol, we have Since p ≡  mod , from [] (see formula () of Section .) we know that From () and () we may immediately deduce formula (I) of Lemma . Now we prove formula (II) of Lemma . It is clear that from (I) we have where I(p) = -, if p = k + ; and I(p) = , if p = k + .

Combining () and (), we have the identity
This proves formula (II) of Lemma . Similarly, from (I) and () we can also prove that if p = k + , then we have