scholarly journals SYMMETRIC HOMOGENEOUS DIOPHANTINE EQUATIONS OF ODD DEGREE

2013 ◽  
Vol 09 (04) ◽  
pp. 867-876 ◽  
Author(s):  
M. A. REYNYA
Keyword(s):  
Diophantine Equations ◽  
Parametric Solution ◽  
Elementary Approach ◽  
Symmetric Form ◽  
Model Case ◽  
Parametric Solutions ◽  
Odd Degree

An elementary approach for finding non-trivial parametric solutions to homogeneous symmetric Diophantine equations of odd degree in sufficiently many variables is presented. The method is based on studying a model case of quintic symmetric Diophantine equations in six variables. We prove that every symmetric form of odd degree n ≥ 5 in 6 ⋅ 2n - 5 variables has a rational parametric solution depending on 2n-8 parameters. We also present a solution to a problem of Waring type: if F(x1,…, xN) is a symmetric form of odd degree n ≥ 5 in N = 6 ⋅ 2n-4 variables, then for any q ∈ ℚ the equation F(xi) = q has a rational parametric solution depending on 2n - 6 parameters.

Representations of a rational number as a sum of odd powers

2016 ◽  
Vol 12 (04) ◽  
pp. 903-911
Author(s):  
M. A. Reynya
Keyword(s):  
Number Theory ◽  
Rational Number ◽  
Diophantine Equations ◽  
Symmetric Form ◽  
Parametric Solutions ◽  
Odd Degree

In this paper, we substantially generalize one of the results obtained in our earlier paper [M. A. Reynya, Symmetric homogeneous Diophantine equations of odd degree, Int. J. Number Theory 28 (2013) 867–879]. We present a solution to a problem of Waring type: if [Formula: see text] is a symmetric form of odd degree [Formula: see text] in [Formula: see text] variables, then for any [Formula: see text], [Formula: see text], the equation [Formula: see text] has rational parametric solutions, that depend on [Formula: see text] parameters.

Development of the mathematic model of dissymmetric bigram cryptosystem based on parametric solution family of multi-degree system of Diophantine equations

2020 ◽  
Author(s):  
V. O. Osipyan
Keyword(s):  
Diophantine Equations ◽  
Mathematic Model ◽  
Parametric Solution

Предложен новый подход разработки дисимметричной биграммной криптосистемы (ДБК) на основе многопараметрических решений многостепенных систем диофантовых уравнений (МСДУ), обобщающий принцип построения криптосистем с открытым ключом. Вводится новое понятие равносильности числовых наборов или параметров заданной размерности и порядка. Описанные математические модели СЗИ демонстрируют потенциал применения МСДУ для разработки СЗИ с высокой степенью надёжности

scholarly journals Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2779
Author(s):  
Petr Karlovsky
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Natural Numbers ◽  
Parametric Solutions ◽  
Special Cases ◽  
Positive Integers

Diophantine equations ∏i=1nxi=F∑i=1nxi with xi,F∈ℤ+ associate the products and sums of n natural numbers. Only special cases have been studied so far. Here, we provide new parametric solutions depending on F and the divisors of F or F2. One of these solutions shows that the equation of any degree with any F is solvable. For n = 2, exactly two solutions exist if and only if F is a prime. These solutions are (2F,2F) and (F + 1, F(F + 1)). We generalize an upper bound for the sum of solution terms from n = 3 established by Crilly and Fletcher in 2015 to any n to be F+1F+n−1 and determine a lower bound to be nnFn−1. Confining the solutions to n-tuples consisting of distinct terms, equations of the 4th degree with any F are solvable but equations of the 5th to 9th degree are not. An upper bound for the sum of terms of distinct-term solutions is conjectured to be F+1F+n−2n−1!/2+1/n−2!. The conjecture is supported by computation, which also indicates that the upper bound equals the largest sum of solution terms if and only if F=n+k−2n−2!−1, k∈ℤ+. Computation provides further insights into the relationships between F and the sum of terms of distinct-term solutions.

scholarly journals Some diophantine problems concerning equal sums of integers and their cubes

2010 ◽  
Vol Volume 33 ◽  
Author(s):  
Ajai Choudhry
Keyword(s):  
Symmetric Solution ◽  
Diophantine Equations ◽  
Simple Solution ◽  
Parametric Solution ◽  
Quadratic Polynomials ◽  
Simultaneous Diophantine Equations ◽  
International Audience ◽  
Diophantine Problems ◽  
The Given ◽  
Complete Solutions

International audience This paper gives a complete four-parameter solution of the simultaneous diophantine equations $x+y+z=u+v+w, x^3+y^3+z^3=u^3+v^3+w^3,$ in terms of quadratic polynomials in which each parameter occurs only in the first degree. This solution is much simpler than the complete solutions of these equations published earlier. This simple solution is used to obtain solutions of several related diophantine problems. For instance, the paper gives a parametric solution of the arbitrarily long simultaneous diophantine chains of the type $x^k_1+y^k_1+z^k_1=x^k_2+y^k_2+z^k_2=\ldots=x^k_n+y^k_n+z^k_n=\ldots,~~k=1,3.$ Further, the complete ideal symmetric solution of the Tarry-Escott problem of degree $4$ is obtained, and it is also shown that any arbitrarily given integer can be expressed as the sum of four distinct nonzero integers such that the sum of the cubes of these four integers is equal to the cube of the given integer.

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