https://tcs.nju.edu.cn/wiki/api.php?action=feedcontributions&feedformat=atom&user=98.20.210.218 2026-05-01T09:48:06Z User contributions MediaWiki 1.45.3 https://tcs.nju.edu.cn/wiki/index.php?title=Diophantine_equation&diff=7862 2014-12-25T03:24:21Z

<p>98.20.210.218: </p> <hr /> <div>A '''Diophantine equation''' is an [[equation]] that only takes [[integer]] coefficients, and that can be written as &lt;math&gt;f(x_1, x_2, x_3, . . ., x_n) = 0&lt;/math&gt;, where ''f'' is a [[polynomial]]. '''Diophantine analysis''' is a branch of [[mathematical analysis]], concerned with such equations. Typical questions include:<br /> #Are there any solutions?<br /> #Are there any solutions beyond some that are easily found by inspection?<br /> #Are there finitely or infinitely many solutions?<br /> #Can all solutions be found in theory?<br /> #Can one in practice compute a full list of solutions?<br /> Very often such problems were unresolved for centuries. With time, mathematicians came to understand their depth (in some cases), rather than treat them as puzzles. The equation is named after [[Diophantus of Alexandria]], a mathematician, who described them first. The restriction on integer coefficients makes sense, when one is concerned about finding [[Divisor]]s, or in the case of [[modular arithmetic]]. In everyday life, many equations solve problems where only whole numbers make sense: A product is composed of many parts, but only whole pieces can be produced. <br /> <br /> Some diophantine equations are very famous. These include the [[Pythagorean triple]], [[Fermat's Last Theorem]] and [[Pell's equation]]. [[Hilbert's problems|Hilbert's tenth problem]] was to find an [[algorithm]] to decide, whether a given Diophantine equation has an integer solution. <br /> <br /> {{math-stub}}<br /> <br /> [[Category:Equations]]</div>

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