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As mathematicians continue to study mathematics, often times they run into a problem which takes a considerable amount of effort to solve. For instance, trying to factor polynomials has lead to a large portion of group theory and Galois theory especially. Another example, which I have heard reference from one of my professors, is that the study of elliptic functions has lead to a large amount of Algebraic Geometry. The original goal being how to integrate elliptic curves turned into quite an interesting field over the years.

My question, although "broad", is this:

What are some mathematical problems that have been forgotten?

In the process of trying to solve one question, often times many more questions are discovered. I find it hard to believe that no question has been forgotten. As in, sometimes the cutting-edge work of mathematics has trumped the questions that seemed of lesser importance at the time of research, and after a while most everyone has forgotten about them. Surely some of us must remember! Or, are there any stories of such problems like this that have sparked new mathematical research over the years?

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    $\begingroup$ I used to know that, but I don't remember anymore. $\endgroup$
    – Asaf Karagila
    Commented Dec 23, 2014 at 6:49
  • $\begingroup$ I was going to say just that! $\endgroup$ Commented Dec 23, 2014 at 7:25
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    $\begingroup$ I'm not sure this is what you are looking for but the study of syllogisms dominated the study of logic for centuries. Syllogisms are not yet completely forgotten, but are considered entirely unimportant. $\endgroup$
    – MJD
    Commented Dec 23, 2014 at 7:26
  • $\begingroup$ @MJD: Can you give references to your claim? $\endgroup$
    – user170039
    Commented Jun 22, 2016 at 13:54

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The n-body problem dates back to the ancient Greeks and was once considered the key to understanding the movement of the planets, and, by extension, the very nature of the Universe.

A "geometrical" (i.e. exact) solution was always hoped for and expected, but the desire to compute orbital motions in practice led to the development of numerical methods; in the meanwhile, it was discovered that the case $n>2$ is chaotic (an early motivation for chaos theory).

Both numerical analysis and chaos theory have since dwarfed the original problem in terms of the amount of research effort dedicated to them, and while it's difficult to say that the n-body problem is "forgotten" (if it were, we couldn't be discussing it) I'm not aware of any substantial theoretical work (let alone progress) done on it for several decades now beyond what is already covered by the more general field of numerical methods.

Add to this the fact that the relativistic version of the n-body problem is fundamentally different from the classical formulation, leading to a solution for the classical problem being of much lesser interest in today's context of large-scale celestial mechanics than it would have been two centuries ago, and you have this once monumental problem being now reduced almost to its historical importance alone.

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    $\begingroup$ Maybe you're not aware of it, but there has been a lot of recent interesting work on the classical $n$-body problem, e.g. the discovery of "choreographies". See scholarpedia.org/article/Computational_celestial_mechanics and references there, also maths.manchester.ac.uk/~jm/Choreographies $\endgroup$ Commented Dec 23, 2014 at 7:41
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    $\begingroup$ @RobertIsrael: Another interesting (fairly) recent development is Xia's 1992 paper proving the existence of non-collision singularities, where one particle is ejected to infinity in finite time; this settled an old question by Painlevé. $\endgroup$ Commented Dec 23, 2014 at 10:33
  • $\begingroup$ Maybe of much lesser interest, but hard to say it has been forgotten. $\endgroup$
    – J.-E. Pin
    Commented Sep 20, 2020 at 4:23
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I think that one of the problems that has sparked a great deal of mathematical research over the centuries could be Pappus' problem. In its classical formulation, we are given 3 lines: $l_{1} \ldots l_{3}$, and three angles: ${\theta}_{1} \ldots {\theta}_{3}$. We then define, for any point $P$ on the plane, a distance $d_{i}$ as the length of the oblique segment drawn from $P$ towards a line $l_{i}$ at an angle $\theta_{i}$. The task is to find the locus of the points $P$ whose distances $d_{1} \ldots d_{3}$ from the given lines $l_{1} \ldots l_{3}$ have a constant ratio $d_{1}^2:d_{2}d_{3}$. The problem originates in Greek mathematics although, as Pappus recounts in the Mathematical Collection (here is an overview of Pappus' life and work), it was a stumbling block for ancient geometers as soon as the number of given lines and angles exceeded the case $n=3$ (the "product" of more than three segments had no meaning for Greek geometers). The problem was solved by Descartes, in his Géométrie (1637), for an arbitrary number of given lines and angles. Descartes' comprehensive solution relies on the introduction of the now familiar, but revolutionary at the time, representation of curves via algebraic equations in two unknowns (an overview can be found here). Pappus' problem was one of the main driving motives behind Descartes' idea to represent curves via equations, and sparked the birth of analytic geometry with its long-lasting research tradition.

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One category of mathematics that languished for many centuries is neusis construction. It was often used by mathematicians, such as Archimedes, in ancient times to solve problems that had no known solution via Euclidean methods. Presumably such constructions were pending the development of a Euclidean solution. But the method was deemed so inferior and incompatible with Euclid's axioms that it lost its appeal for millennia. The emergence, in the 19th century, of proofs that Euclidean construction is very limited in scope has made neusis construction more interesting again, as with Benjamin and Snyder's constructibility proof for the regular 11-gon. The long hiatus of neusis construction is one reason why, even today, we do not fully know its scope.

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I suggest that you look up "unsolved" problems in the atlas of unsolved problems posted online. Another way to find "them" is to find your area first, then look up top mathematicians pages and in there you can find some post about problems they are still having trouble resolve.

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  • $\begingroup$ you mean they forgot/given up their intention to solve? $\endgroup$
    – Narasimham
    Commented Aug 31, 2015 at 19:37

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