The main purpose of this paper is to use analytic methods and properties of quartic Gauss sums to study a special fourth power mean of a two-term exponential sums modp, with p an odd prime, and prove interesting new identities. As an application of our results, we also obtain a sharp asymptotic formula for the fourth power mean.

The Girard and Waring formula and mathematical induction are used to study a problem involving the sums of powers of Fibonacci polynomials in this paper, and we give it interesting divisible properties. As an application of our result, we also prove a generalized conclusion proposed by R. S. Melham.

The main purpose of this paper is to use the mathematical induction and the properties of Lucas polynomials to study the power sum problem of Lucas polynomials. In the end, we obtain an interesting divisible property.

The main purpose of this paper is applying the analysis method, the properties of Lucas polynomials and Gauss sums to study the estimation problems of some kind hybrid character sums. In the end, we obtain several sharp upper bound estimates for them. As some applications, we prove some new and interesting combinatorial identities.

We consider any prime number p . Let k , s be two positive integers. We are interested in the arithmetic progressions (sequences) with the common difference s and length k , where the sequence entries are from the set of quadratic residue modulo p or the set of quadratic nonresidue modulo p . The numbers of such sequences are denoted as N p k , s and N p ′ k , s , respectively. In this paper, we apply analytic number theory methods, in particular, properties of Legendre’s symbol modulo p and character sums, to study the numbers N p k , s and N p ′ k , s . Exact formulas are given for certain values of k and s under some restrictions. In addition, estimation formulas in other cases are given.

The aim of this study is to use analytic methods and the properties of character sums to calculate the number of solutions to certain congruence equations modulo an odd prime. Formulas for the exact number of solutions are derived.