It has been proved that there is no algorithm to solve Diophantine equations, for that reason I want to know what are the Diophantine equations that physicists or chemists need to solve? Or any other still unsolved Diophantine equations that have applications in other area of science.
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$\begingroup$ welcome to SE! in terms of applications of diophantine equations, you might want to check out "integer programming" (en.wikipedia.org/wiki/Integer_programming). It is very common in practical applications to have to solve optimization problems where the inputs are integers (as opposed to just real numbers). Integer programming is a very deep / active area of research. $\endgroup$– Adithya ChakravarthyCommented May 16, 2022 at 21:23
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$\begingroup$ The proof of unsolvability effectively says that Diophantine equations are equivalent to Turing machines, so there's a sense in which this question doesn't have a sensible answer. $\endgroup$– Peter TaylorCommented May 16, 2022 at 23:35
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$\begingroup$ During my attempts to search and look at Diophantine equations, I found that there are some equations that are solved, but they have no use, either within mathematics or in other applied fields such as physics. I think it is better to identify the problems that need to be solved by a particular type of Diophantine equations. $\endgroup$– user482376Commented May 16, 2022 at 23:42
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$\begingroup$ If you want to know what Diophantine equations physicists and/or chemists need to solve, shouldn't you be asking physicists and/or chemists, not mathematicians? $\endgroup$– Gerry MyersonCommented May 17, 2022 at 7:42
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$\begingroup$ Okay, thank you so much $\endgroup$– user482376Commented May 17, 2022 at 8:04
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1 Answer
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Pythagorean triples can be used to construct right angles and so had applications to surveying in the ancient world, see this link.
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$\begingroup$ Thank you for your answer but my quation is I want to know what are the problem in physics or chemistry or any other problem need to solve some type of Diophantine equations because there are many Diophantine equations that have solved but they haven't any uses in the other science like physics . I mean that we focus in solving the diophantine equations that other have some applications not any equations $\endgroup$– user482376Commented May 16, 2022 at 21:09
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$\begingroup$ My answer was somewhat tongue-in-cheek. Diophantine equations are the essence of pure mathematics.. $\endgroup$– StoppleCommented May 16, 2022 at 22:49