Diophantine equations

Constitution of a Chemical Reaction and Reaction Balancing

Published in John Andraos, Reaction Green Metrics, 2018

A Diophantine equation is a polynomial equation in two or more independent variables having only integer solutions. It is named in honor of the third-century Greek mathematician Diophantus of Alexandria. For our purposes, cases where multiple sets of stoichiometric coefficients are applicable to balancing a redox reaction are examples of simple linear Diophantine equations of the form ax+by=c.

Analytical Methods

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Published in Alexander D. Poularikas, Handbook of Formulas and Tables for Signal Processing, 2018

Alexander D. Poularikas

A Diophantine equation has integer coefficients and integer solutions. As an example, the equation ax + by = c, a,b,c Z (a), has integer solutions x and y if and only if (a,b) divides c. In particular, ax + by = 1 is solvable (a,b) = 1. If xo,yo is a particular solution of (a), then the general solution is x = xo + nb/(a,b), y = yo - na/(a,b), n Z.

Fixed-point iterative approach for solving linear Diophantine systems with bounds on the variables

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Published in Cyber-Physical Systems, 2022

Haocheng Yu, Luyao Yang, Jinyu Dai, Baoping Jiang, Zhengtian Wu, Shuxian Zhu

Systems of linear Diophantine equations can be applied to various situations, such as in solving the market split problem and Frobenius problem. Consider the following linear Diophantine equations:

Diophantine equations

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Constitution of a Chemical Reaction and Reaction Balancing

Analytical Methods

Fixed-point iterative approach for solving linear Diophantine systems with bounds on the variables