Real-world applications of number theory

Autor: Rahim Rahmati-Asghar
Jazyk: perština
Rok vydání: 2024
Předmět:
Zdroj: ریاضی و جامعه, Vol 8, Iss 4, Pp 93-104 (2024)
Druh dokumentu: article
ISSN: 2345-6493
2345-6507
DOI: 10.22108/msci.2023.136490.1555
Popis: The above abstract has been extracted by the translator from the original article (J. Klaška, Real-world applications of number theory, South Bohemia Mathematical Letters, 25 no. 1 (2017) 39–47.) The present paper is concerned with practical applications of the number theory and is intended for all readers interested in applied mathematics. Using examples we show how human creativity can change the results of the pure mathematics into a practical usable form. Some historical notes are also included. 1. IntroductionGerman mathematician Johann Carl Friedrich Gauss (30 April 1777-23 February 1855), regarded as one of the greatest mathematicians of all time, claimed: "Mathematics is the queen of the sciences and number theory is the queen of mathematics." However, for many years number theory had only few practical applications. It is well known that the great English number theorist Godfrey Harold Hardy (7 February 1877-1 December 1947) believed that number theory had no practical applications. See his essay "A Mathematician's Apology" [16]. Over the 20th and 21st centuries, this situation has changed significantly. Contrary to Hardy's opinion, many practical and interesting applications of number theory have been discovered. The present paper brings some remarkable examples of number theory applications in the real world. The paper can be regarded as a loose continuation of the author's preceding work [19] and [20]. Let $n$ be a positive integer, $\geq 2$. Then, the equation (1.1)$$a_1x_1+\ldots+a_nx_n=m$$is said to be a linear Diophantine equation if all unknowns $x_1,\ldots,x_n$ and all coefficients $a_1,\ldots,a_n,m$ are integers. For general methods for solving (1.1), see for example [5], [25], and [24, pp. 27-31]. In the following we give some interesting examples of using Diophantine equations in the natural sciences. 2. The resultsAs the first example we show some applications of a linear Diophantine equation to problems in chemistry. In particular, we will deal with the balancing of chemical equations. See [6]. Consider a chemical equation written in the form (2.1)$$x_1A_{a_1}B_{b_1}C_{c_1}\ldots +x_2A_{a_2}B_{b_2}C_{c_3}\cdots +\cdots\rightarrow x'_1A'_{a'_1}B'_{b'_1}C'_{c'_1}\cdots +x'_2A'_{a'_2}B'_{b'_2}C'_{c'_3}\cdots +\cdots$$where $A, B, C,\ldots$ are the elements occurring in the reaction, $a_1, b_1, c_1,\ldots,a'_1, b'_1, c'_1,\ldots$ are positive integers or $0$, and $x_1, x_2,\ldots,x'_1, x'_2,\ldots$ are the unknown coeffcients of the reactants and products. Then, we hav (2.2)\begin{array}{rcl}x_1a_1+x_2a_2+\cdots&=&x'_1a'_1+x'_2a'_2+\cdots\\x_1b_1+x_2b_2+\cdots&=&x'_1b'_1+x'_2b'_2+\cdots\\x_1c_1+x_2c_2+\cdots&=&x'_1c'_1+x'_2c'_2+\cdots\\&\cdots&\end{array} Clearly, each equation of (2.2) expresses the law of conservation of the number of atoms for any particular element $A,B,C,\ldots$. Finding all integer solutions $[x_1,x_2,\ldots,x'_1,x'_2,\ldots]$ of (2.2) is a nice elementary problem of Diophantine analysis. In the second example we show how linear Diophantine equations can be used to determine the molecular formula [6]. Assume that a substance with a molecular weight of $m$ contains elements $A, B, C,\ldots$ with atomic weights $a, b, c,\ldots$ and that $x,y,z,\ldots$ represent the numbers of atoms of $A, B, C,\ldots$ in a molecule. Then, we have (2.3)$$ax+by+cz+\ldots=m.$$Let $\alpha,\beta,\gamma,\ldots$ denote the integers nearest the values $a, b, c,\ldots$ and $\mu$ denote theinteger nearest m. Then, (2.3) can be replaced by the linear Diophantine equation (2.4)$$\alpha x+\beta y+\gamma z+\cdots=\mu.$$ If we require that the values $x, y, z,\ldots$ in (2.4) should be reasonably small, we can solve (2.4) under a condition (2.5)$$-\frac{1}{2}
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