This paper introduces the concept of rational quadratic residues, namely fractional quadratic residues. It also presents a method for determining quadratic residues with fractions as moduli, as well as computational symbols for rational quadratic residues and formulas for converting rational Legendre symbols to general Legendre symbols. Formulas for calculating the number of rational quadratic residues and rational quadratic non-residues are derived. Additionally, several practical application e...

Integral Equation  Number Theory 

In this paper, a new concept is proposed: semi-primitive root. The basic theory system of semi-primal roots is established, and the congruence equations and propositions of indefinite equations are solved by the theory of semi-primal roots. The security problem of using the same value to digitally sign multiple different information in discrete logarithm encryption is solved.

Mathematics  Integral Equation  Number Theory 

In this paper, the multiway extension of the Euler proposition is presented. By means of inductive reasoning, it is proved that there are positive integer solutions to five kind of indefinite equations, and the uniqueness of the solution of the indefinite equations in these propositions is demonstrated. Three related conjectures are proposed. Finally, the general method to understand these indeterminate equations is pointed out. 

Integral Equation  Number Theory 

1) Fermat has proved that x⁴ y⁴=z²   has no positive integer solution, and in 2011, J. Cullen [1] reported that x,y,∈{0,1,...,10⁷}, x⁴ y⁴ 1 is not a square greater than 1, and conjecture:x⁴ y⁴ 1≠z²,z∈{2,3,...},x,y,∈{0,1,...}. On May 15, 2021, Sun Zhiwei [2] proposed that neither x⁴ y⁴ 1(x,y,∈N) is a perfect power based on Cullen’s conj...

Integral Equation  Number Theory 

In this paper, the concepts of combination number, factorial and Euler function are generalized, and the concepts of generalized combination number, generalized Euler function and m factorial of n are proposed. Several typical Identities and congruence of generalized combination number are proved, and the calculation formulas of generalized Euler function are derived. Using the m factorial of n, the congruence formula of the modular prime of the original facto...

Integral Equation  Number Theory 

In this paper, a new mathematical method is used to study the indefinite equations of binary quadratic and binary arbitrary order, the problem of judging and solving these indefinite equations with or without solutions is solved.

Integral Equation  Number Theory 

With the help of Mindlin’s tensor, the problem of heterogeneity in an elastic half-space comes down to the solution system of integrable equations. Different versions of the solution of such systems are considered. The main attention is paid to the solution of integrable equations with small parameters.

Integral Equation 

If to apply bidimensional Fourier’s transform to homogeneous system of equations of the theory of elasticity, then we will receive system of ordinary differential equations. The general solution of this system contains 6 arbitrary constants and allows to solve problems for the layer and the multilayer environment. It is shown that it is convenient to do statement and the solution of such tasks in the matrix form. The task for the layer on the elastic half-space is solved. Ways of inverse of Four...

Integral Equation 

The purpose of this paper is to study the stability of nonlinear fractional Duffing equation where , by analysing the eigenvalues generated from the system of the given differential equation. Graphical results furthermore show the effect of the damping and nonlinear parameter on the system. Our contribution relies on its application to the choice of hard/soft spring in the mechanism of shock absorbers.

Integral Equation  Ordinary Differential Equation 

The main aim in this work is to obtain an integral inequality with a clear estimate on time scales. The obtained inequality is used as a tool to investigate some basic qualitative properties of solutions to certain nonlinear Volterra-Fredholm integrodifferential equations on time scales.

Integral Equation