Research On A Mean Value Estimate For A Special Gauss Sums And Several Diophantine Equation Problems

The main purpose of this dissertation is to study some special problems involv-ing the mean value estimate on the famous Gauss sums, D. H. Lehmer problem, the integer points of elliptic curve, the solvability of exponential Diophantine sys-tem and several other Diophantine equations, which play important roles in analytic number theory and Diophantine equation. That is, the estimate problem of a spe-cial Gauss sums is studied by using analytic methods. The integer points problem of two kinds of elliptic curves, the divisibility problem on a kind of exponential Diophantine system and three kinds of Diophantine equations are discussed. Some meaningful results are given. The main achievements contained in this dissertation can be described as follows:Chapter1presents the introduction of number theory, the research background and introduction of analytic number and Diophantine equation respectively. Then the main works of this dissertation are introduced.In chapter2, a special Gauss sums is studied by using the analytic method and the properties of general Kloosterman sums and combining the D. H. Lehmer problem, and a sharp upper bound estimate is given.In chapter3, two different kinds of the integer points problems of elliptic curve are studied. Firstly, The integer point of the general elliptic curve y2=x3+(36n2-9)x-2(36n2-5) is given and proved by combine some properties of quadratic and quartic Diophantine equations with elementary analysis. Then, the integer point elliptic curve y2=px(x2+1) is discussed by using the elementary analysis. Meanwhile, two criterions for the elliptic curve which has positive integer points are given.In the fourth chapter, the solvability of the exponential Diophantine system2x+py=qz and p+2=q is studied by using the algebraic and elementary methods. The problem of the exponential Diophantine system is solved completely. The only solution of this system is given and proved. The fifth chapter discuss the solvability of three kinds of Diophantine equation. Firstly, the solvability of the equation υk=s2±1is discussed, and its all positive integer solution (k, s) are completely determined. Secondly, the properties of prime divisors of odd perfect numbers are discussed by using some results on higher degree Diophantine equations. Then a result for an odd perfect numbers is improved and proved. Finally, the solvability of two binary quadratic Diophantine equation x2-Dy2=±2are discussed and a necessary and sufficient condition for them are given and proved.In the sixth chapter, the coefficient problem of the trinomials f(X)=Xn-BX+A has an irreducible quadratic factor is discussed by using a lower bound for two logarithms in the complex case. The bound for coefficients of the trinomials is given and proved. A similar result for Xn-BXk+A can be obtained by the above result and Using the divisibility of Lucas numbers.