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The question that was asked compares Diophantine equations to differential equations, with the famous differential equations first arising due to physical arguments before taking on a life of their own. The interests of mathematicians long ago in simple questions about geometry or powers of numbers are what gave rise to the classical Diophantine equations. That such equations still have interest is due to them "taking on a life of their own": connections are found with important themes of mainstream mathematics, so those old equations become good examples of advanced theories.

For example, special instances of Pell's equation x^2-dy^2=1 occurred in the work of Greek and Indian mathematicians thousands of years ago. One reason is related to irrationality. Since sqrt(2) is irrational, x^2 - 2y^2 is not 0 when the variables are positive integers and you might ask, particularly in those old days when there was not very advanced math, what the smallest nonzero integral value of x^2-2y^2 could be, and how such values occur. This leads to x^2-2y^2 = +1 or -1, and both equations have many integral solutions by a recursive method, as the Indians knew. If you look at x^2 - 3y^2 = +1 or -1 you find quickly that there's no solution when the right side is -1, so already something new happens.

Another way Pell's equation arises is through questions about polygonal numbers, which were a topic of interest long ago. (I am not going to argue that they have some over-arching signifiance today, but what do you expect people back then to have been thinking about?) Since 36 = 6^2 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 is both a square number and a triangular number, you can ask whether there are square-triangular numbers beyond 36, and this is essentially the same as solving an instance of Pell's equation.

Many geometric questions about triangles, esp. right triangles, with integral or rational side lengths lead to low-degree Diophantine equations. The equation a^2 + b^2 = c^2 is too famous to say anything about. Fermat was inspired to show x^4 + y^4 = z^2 has no nontrivial integral solutions in order to prove no right triangle with rational side lengths can have area equal to a perfect square (you can't "square" a rational right triangle). As an unplanned consequence of solving that problem, Fermat had shown the Fermat equation with exponent 4 has no nontrivial integral solutions (replace z with z^2 in the previous equation). The method discovered for by Fermat for this problem was his technique of infinite descent, which he was able to use successfully on other problems, including those with a more positive character (i.e., showing some equation has an integral solution, like primes p = 1 mod 4 being a sum of two squares).

The link found later between Pell's equation and unit groups in quadratic rings provided a reason for number theorists to have a permanent conceptual interest in that equation.

Once we get tired of squares all the time, we might look at squares and cubes. The progressions of pefect perfect squares and cubes keep interlacing and you might ask how close they can come (other than the silly case when they coincide, like with 64 = 8^2 = 4^3). This leads to y^2 = x^3 + 1, y^2 = x^3 - 1, y^2 = x^3 + 2, y^2 = x^3 - 2. Already here we see a very different situation compared to Pell's equation, since these equations will have only a finite number of integral solutions; the case of y^2 = x^3 - 2 is a famous example Fermat used to challenge the British mathematicians. We don't know how Fermat showed the only solutions are (3,5) and (3,-5), but Euler discovered later that prime factorization in the ring Z[sqrt(-2)] gave gives a natural explanation of the result. This was one of the earliest instances of using algebraic integers to solve Diophantine equations in ordinary integers, and still provides a good example for an algebraic number theory course.

Euler looked at y^2 = x^3 + 1 and y^2 = x^3 - 1 and found a way to apply Fermat's idea of infinite descent to show the only integral solutions are the small ones you can find by hand. In the early 20th century Mordell pushed the method of descent further to prove the Mordell part of the Mordell--Weil theorem. Through the influence of Weil and others, the method of descent remains an important tool, although in a language that looks nothing like what Fermat used.

Mordell spent many years of his life studying integral solutions of the equation y^2 = x^3 + k, where k is a fixed nonzero integer. The equation could be justified as having interest because it's one of the simplest examples of an elliptic curve, but it's important for a better reason. The abc-conjecture, which is one of the big open questions in number theory that has connections to many other unsolved problems, does not at first look like it is about Mordell's equation. However, the abc-conjecture abc conjecture turns out to be logically equivalent to specific upper bounds on an integral solution (x,y) to Mordell's equation in terms of the parameter k. So, as Barry Mazur once remarked, the Mordell equation is a far more central topic to all of number theory than its rather special appearance suggests.

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The question that was asked compares Diophantine equations to differential equations, with the famous differential equations first arising due to physical arguments before taking on a life of their own. The interests of mathematicians long ago in simple questions about geometry or powers of numbers are what gave rise to the classical Diophantine equations. That such equations still have interest is due to them "taking on a life of their own": connections are found with important themes of mainstream mathematics, so those old equations become good examples of advanced theories.

For example, special instances of Pell's equation x^2-dy^2=1 occurred in the work of Greek and Indian mathematicians thousands of years ago. One reason is related to irrationality. Since sqrt(2) is irrational, x^2 - 2y^2 is not 0 when the variables are positive integers and you might ask, particularly in those old days when there was not very advanced math, what the smallest nonzero integral value of x^2-2y^2 could be, and how such values occur. This leads to x^2-2y^2 = +1 or -1, and both equations have many integral solutions by a recursive method, as the Indians knew. If you look at x^2 - 3y^2 = +1 or -1 you find quickly that there's no solution when the right side is -1, so already something new happens.

Another way Pell's equation arises is through questions about polygonal numbers, which were a topic of interest long ago. (I am not going to argue that they have some over-arching signifiance today, but what do you expect people back then to have been thinking about?) Since 36 = 6^2 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 is both a square number and a triangular number, you can ask whether there are square-triangular numbers beyond 36, and this is essentially the same as solving an instance of Pell's equation.

Many geometric questions about triangles, esp. right triangles, with integral or rational side lengths lead to low-degree Diophantine equations. The equation a^2 + b^2 = c^2 is too famous to say anything about. Fermat was inspired to show x^4 + y^4 = z^2 has no nontrivial integral solutions in order to prove no right triangle with rational side lengths can have area equal to a perfect square (you can't "square" a rational right triangle). As an unplanned consequence of solving that problem, Fermat had shown the Fermat equation with exponent 4 has no nontrivial integral solutions (replace z with z^2 in the previous equation). The method discovered for Fermat for this problem was his technique of infinite descent, which he was able to use successfully on other problems, including those with a more positive character (i.e., showing some equation has an integral solution, like primes p = 1 mod 4 being a sum of two squares).

The link found later between Pell's equation and unit groups in quadratic rings provided a reason for number theorists to have a permanent conceptual interest in that equation.

Once we get tired of squares all the time, we might look at squares and cubes. The progressions of pefect squares and cubes keep interlacing and you might ask how close they can come (other than the silly case when they coincide, like with 64 = 8^2 = 4^3). This leads to y^2 = x^3 + 1, y^2 = x^3 - 1, y^2 = x^3 + 2, y^2 = x^3 - 2. Already here we see a very different situation compared to Pell's equation, since these equations will have only a finite number of integral solutions; the case of y^2 = x^3 - 2 is a famous example Fermat used to challenge the British mathematicians. We don't know how Fermat showed the only solutions are (3,5) and (3,-5), but Euler discovered later that prime factorization in the ring Z[sqrt(-2)] gave a natural explanation of the result. This was one of the earliest instances of using algebraic integers to solve Diophantine equations in ordinary integers, and still provides a good example for an algebraic number theory course.

Euler looked at y^2 = x^3 + 1 and y^2 = x^3 - 1 and found a way to apply Fermat's idea of infinite descent to show the only integral solutions are the small ones you can find by hand. In the early 20th century Mordell pushed the method of descent further to prove the Mordell part of the Mordell--Weil theorem. Through the influence of Weil and others, the method of descent remains an important tool, although in a language that looks nothing like what Fermat used.

Mordell spent many years of his life studying integral solutions of the equation y^2 = x^3 + k, where k is a fixed nonzero integer. The equation could be justified as having interest because it's one of the simplest examples of an elliptic curve, but it's important for a better reason. The abc-conjecture, which is one of the big open questions in number theory that has connections to many other unsolved problems, does not at first look like it is about Mordell's equation. However, the abc-conjecture turns out to be logically equivalent to specific upper bounds on an integral solution (x,y) to Mordell's equation in terms of the parameter k. So, as Barry Mazur once remarked, the Mordell equation is a far more central topic to all of number theory than its rather special appearance suggests.