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Jun 16, 2025Open Access
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...
Apr 24, 2025Open Access
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.
Feb 26, 2025Open Access
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.
Oct 28, 2024Open Access
Soon, the edition will have the challenge of publishing the prompts of the authors. The theorem of Thales and the theorem of Pythagoras do not escape the rule, and they seem to have been preceded by several millennia. Asking the true paternity of those geometrical realities makes it possible to show that a * b = R2 according to the terms defined by the Theorem of the Circle or Theorem of Apollo stating the constant product of two segments of perpendicular tangents for a...
Sep 27, 2024Open Access
1) Fermat has proved that x 4 y 4=z 2 has no positive integer solution, and in 2011, J. Cullen [1] reported that x,y,∈{0,1,...,10 7}, x 4 y 4 1 is not a square greater than 1, and conjecture:x 4 y 4 1≠z 2,z∈{2,3,...},x,y,∈{0,1,...}. On May 15, 2021, Sun Zhiwei [2] proposed that neither x 4 y 4 1(x,y,∈ N) is a perfect power based on Cullen’s conj...
Aug 28, 2024Open Access
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...
Aug 26, 2024Open Access
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.
Dec 19, 2023Open Access
According to Bertrand’s postulate, we have P n P n≥P n 1. Is it true that for all n>1 then P n-1 P n≥P n 1? Then P n P n-i>P n j where n≥N, N is a large enough value and i, j are natural numbers?
Feb 23, 2023Open Access
This paper attempts to express imaginary number with chemical nucleotide bases (A T, G, C and U) as regards to Quantum Perspective Model. At first, the exact Euler’s formula was written just like as in here “e xi = cos(x) i·sin(x)”. Secondly, twin pi is substituted for x(x = 2Π). Thirdly, the result of this process equals to “e 2iπ = cos(2Π) i·sin(2Π)”. Fourthly, sort up this formula just like as in here: “e 2iπ = 1 i0” = 1. Fifthly, multiply both sides of th...
Mar 30, 2022Open Access
One of the irrational numbers is the square root of ten number. This article researches whether there is a link between the square root of ten number and the genetic sequences. At first, the square root digits of the number ten after the comma are summed one by one. Secondly, the result of the addition corresponds to the nucleotide bases. Thirdly the results thus obtained are expressed as nucleotide bases (A, T, C and G). (A) Adenine, (T) Thymine, (C) Cytosine and (G) Guanine. From this point of...
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