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I am looking for ideas that began as small and maybe naïve or weak in some obscure and not very known paper, school or book but at some point in history turned into big powerful tools in research opening new paths or suggesting new ways of thinking maybe somewhere else.

I would like to find examples (with early references of first appearances if possible or available) of really big and powerful ideas nowadays that began in some obscure or small paper maybe in a really innocent way. What I am pursuing with this question is to fix here some examples showing how Mathematics behave like an enormous resonance chamber of ideas where one really small idea in a maybe very far topic can end being a powerful engine after some iterations maybe in a field completely different. I think that this happens much more in mathematics than in other disciplines due to the highly coherent connectedness of our field in comparison with others and it is great that Mathematics in this way give a chance to almost every reasonable idea after maybe some initial time required to mature it in the minds, hands and papers of the correct mathematicians (who do not necessarily have to be the same that first found that idea).

Summarizing, I am looking for ideas, concepts, objects, results (theorems), definitions, proofs or ways of thinking in general that appeared earlier in history (it does not have to be very early but just before the correct way of using the idea came to us) as something very obscure and not looking very useful and that then, after some undetermined amount of time, became a really powerful and deep tool opening new borders and frontiers in some (maybe other) part of the vast landscape of mathematics.

Edit: I really do not understand the aim in closing this question as it is actually at research level. I am clearly asking for tools that developed into modern research topics. I recognize that some answers are not research level answers, but then you should downvote the answer, not the question. I am really surprised by this decision as one of the persons that vote to close suggested it for publication in a place where it is clear that some of the most valuable answers that this question has received would have never occur precisely because the site that this person suggested is not research oriented. I do not imagine people on HSM answering about species or pointfree topology sincerely as these topics are really current research and not history (and I am interested mainly in current research topics). I do not agree with the fact that a limitation in reading understanding of some people can be enough to close a legitimate question, a question that it is worth for us as mathematicians to do and to show to other people that think that mathematics is useful and powerful the day after being published ignoring thus the true way mathematics is done, with its turnabouts and surprises; a discipline where a simple idea has the power to change the field as $0$ did, as the positional systems did, as sheaves did, or as species did. I am really sad for this decision. It is a pity that so many mathematicians regret the actual way in which their field develops, reject to explain and expose this behavior and hide themselves from this kind of questions about the internal development of ideas in mathematics. I challenge all those who voted to close this question as off-topic to look in HSM for any mention about "locale theory" there.

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    $\begingroup$ Sorry, I misread. $\endgroup$ – Francesco Polizzi Aug 14 at 17:11
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    $\begingroup$ I laughed out loud at big ideas that became small. I have had lots of those. $\endgroup$ – Ville Salo Aug 14 at 17:30
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    $\begingroup$ I think the Shimura-Taniyama conjecture fits the bill. At first Serre didn't even consider it to be a natural question and now of course automorphy theorems are an industry in their own right. $\endgroup$ – user145520 Aug 14 at 18:00
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    $\begingroup$ I would remove the tags for philosophy and sociology. The question is indeed general, historical, and soft, but it doesn’t ask about social interactions or anything where philosophical reasoning would seem helpful. $\endgroup$ – Matt F. Aug 14 at 21:47
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    $\begingroup$ @MattF. I appreciate your comment but I think that this question has (maybe in a low level if you feel so) some philosophical and sociological background in the sense that, first, it can be considered as a question about gnoseology and epistemology of ideas within mathematics and how this knowledge is able to resonate and grow inside the mathematical box until it forms a wide highway among the mathematical paths even when the first appearance of the idea looked innocent... $\endgroup$ – Hvjurthuk Aug 14 at 22:15

21 Answers 21

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In a letter to Frobenius, Dedekind made the following curious observation: if we see the multiplication table of a finite group $G$ as a matrix (considering each element of the group as an abstract variable) and take the determinant, then the resulting polynomial factors into a product of $c$ distinct irreducible polynomials, each with multiplicity equal to its degree, where $c$ is the number of conjugacy classes of $G$. This is now known as Frobenius determinant theorem, and it is what led Frobenius to develop the whole representation theory of finite groups (https://en.wikipedia.org/wiki/Frobenius_determinant_theorem).

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    $\begingroup$ Amazing answer! Really good one. This works in a very concrete layer of mathematics and moreover it opened a huge branch of mathematics! $\endgroup$ – Hvjurthuk Aug 15 at 17:19
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    $\begingroup$ In Introduction to Representation Theory by Etingof et al., Gerovitch explains how Frobenius used group determinants with great facility to prove what we would nowadays call fundamental theorems of representation theory. Frobenius's methods have been largely forgotten since they have been replaced by other methods, but I wonder if there remain undiscovered treasures lying in wait for someone who decides to take a fresh look at group determinants. $\endgroup$ – Timothy Chow Aug 17 at 21:25
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The problem of the seven bridges of Königsberg is surely one of the best-known examples of this. Euler apparently didn't even consider this problem to be mathematical when he solved it, but in doing so he introduced the basic concepts of graph theory (a field which did not really begin to take off until a century and a half later).

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    $\begingroup$ ''Euler apparently didn't even consider this problem to be mathematical when he solved it.'' Is there some evidence for this? He noted that in order to move across each bridge only once, the number of bridges adjacent to each vertex must be an even number apart from the start point and the end point, resolving the Konigsberg problem in the negative. This was very simple for him, but surely he must have seen it as a mathematical argument? $\endgroup$ – Hollis Williams Aug 15 at 23:33
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    $\begingroup$ In a letter to the mayor of Danzig, Euler wrote that "this type of solution bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it". $\endgroup$ – lambda Aug 15 at 23:45
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    $\begingroup$ This still seems strange to me though. The result was presented at the St. Petersburg Academy in 1735 and then published in a journal article ''The solution of a problem relating to the geometry of position'' in 1741 so he must have known its significance and explicitly referred to the geometric nature of the problem in the title of the article. Perhaps he considered it more of a scientific problem. $\endgroup$ – Hollis Williams Aug 16 at 0:09
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    $\begingroup$ It's possible that he changed his mind about it. I found the article "The Truth About Konigsberg" by Brian Hopkins and Robin Wilson which includes the quote I gave, but upon reading more closely it seems he wrote that letter before he'd solved the problem, and perhaps felt differently after seeing how it related to Leibniz's proto-topological ideas. It still feels like a good example given that Euler did not to my knowledge do any further work along the same lines and neither did anyone else for some time. $\endgroup$ – lambda Aug 16 at 0:26
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    $\begingroup$ Also, I don't mean to argue, but I'm not convinced either that he did no further work on the same lines (ie. graph theory). In his paper Recherches sur un nouvelle espèce de quarrés magiques' (Researches on a new kind of magic squares)", he studied the problem of moving around a chessboard with a knight and visiting each square only once (known as the problem of the knight's tour). This is basically a Hamiltonian path problem in graph theory. This is not to mention his study of plane mazes, which are basically graphs. $\endgroup$ – Hollis Williams Aug 16 at 19:12
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Cantor's monumental investigation of the infinity started very innocently as a method to understand the uniqueness of the representation of a function by trigonometric series.

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    $\begingroup$ I wonder if there is a book about Cantor that I can read. He was seriously mistreated at the time and even fell ill to depression if I heard correctly. $\endgroup$ – dezdichado Sep 21 at 21:24
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    $\begingroup$ @dezdichado: The standard biography in English is Dauben's biography. A well known biography in German is this book by Purkert and Ilgauds. There are also numerous published papers in many different languages on all phases of Cantor's life and work written throughout the period since his death (I've intentionally left the quantifer order somewhat ambiguous), but either of these two books should be more than enough for someone starting out. $\endgroup$ – Dave L Renfro Sep 22 at 14:52
  • $\begingroup$ @DaveLRenfro I see, thanks! $\endgroup$ – dezdichado Sep 22 at 15:31
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Integration by parts would seem like a good example. Whoever first used it to integrate a function such as $x\exp(x)$ could certainly not have anticipated the fundamental role it would once play in the theory of PDEs.

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    $\begingroup$ "When Peter Lax went to receive the national medal of science, he was asked by the other recipients about his merits. His answer was (apocryph) I integrated by parts." mathoverflow.net/posts/60908/revisions $\endgroup$ – Gerry Myerson Aug 14 at 23:13
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    $\begingroup$ Surely 'once' in "… it would once play …" should be 'eventually' or so (since I assume it's not done playing that role)? $\endgroup$ – LSpice Aug 14 at 23:44
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    $\begingroup$ And the linear algebra viewpoint helps make it clear why integration by parts is so fundamental: integration by parts states that the adjoint of $\frac{d}{dx}$ is $-\frac{d}{dx}$ (or the adjoint of $\nabla$ is $-\text{div}$) in a setting where boundary terms vanish. $\endgroup$ – littleO Aug 15 at 7:39
  • $\begingroup$ Oh, great Peter Lax, always so charismatic! $\endgroup$ – Hvjurthuk Aug 15 at 8:47
  • $\begingroup$ Totally accidental downvote on the phone app in my pocket last night, my apologies. (If you edit so I can change it I’ll upvote) $\endgroup$ – Alec Rhea Aug 15 at 23:11
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Pick's theorem states that the area $A$ of a polygon $P$ in $\mathbb{R}^2$, whose vertices are in the lattice $\mathbb{Z}^2$, can be computed by means of the formula $$A=I+\frac{B}{2}-1,$$ where $I$ is the number of lattice points in the interior of $P$, and $B$ is the number of points in the boundary of the polygon $P$. George Pick published this theorem in 1899, in his article "Geometrisches zur Zahlenlehre" [Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines für Böhmen "Lotos" in Prag. (Neue Folge). 19: 311–319]. Apparently, this result remained unknown until the middle of the 20th century when Hugo Steinhaus included it in his book "Mathematical Snapshots".

This beautiful result is a precursor of theories about "counting lattice points in polyhedra" (e.g., Ehrhart's theory, and generalized Euler-MacLaurin summation formulas) which intersect, as far as I know, with problems in linear programming, values of number theoretic zeta functions, toric varieties, and even physics (I've heard).

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    $\begingroup$ Nice, didn't know that Pick's theorem was mor ethan a simple curiosity! $\endgroup$ – Antoine Labelle Aug 16 at 0:41
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I don't know if this post will answer your question but it's about the higher homotopy groups of topological spaces. The story is told here: https://ncatlab.org/nlab/show/homotopy+group.

In 1932, E. Čech proposed a definition of higher homotopy groups using maps of spheres, but the paper was rejected for the Zurich ICM since it was found that these groups $\pi_n(X,a)$ were abelian for $n \geq 2$, and so do not generalize the fundamental group in the way that was originally desired. Nonetheless, they have proved to be extremely important in homotopy theory, although more difficult to compute in general than homology groups.

The higher homotopy groups of spheres are central in many problems and conjectures in mathematics and they are known to be very difficult to calculate.

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    $\begingroup$ This one is also a great example although being rejected for the ICM is not an obscure beginning at all: it is more like a failed shiny beginning. $\endgroup$ – Hvjurthuk Aug 15 at 8:50
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Pointfree topology originated in some sense in terms of "local lattices" (lokale Strukturen) in a 1957 paper of Charles Ehresmann, but the topic was of little interest until it took off with a 1972 paper of John Isbell who argued that the pointfree approach to topology is in some ways superior to the usual approach. A great overview of the history is given in the entry Elements of the History of Locale Theory by Peter Johnstone in the 3rd volume of the Handbook of the History of General Topology.

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    $\begingroup$ This is actually a really good answer! I imagine that when the branch is more abstract it is even easier to find these examples of ideas that appear at first with low relevance and then suddenly at some point they explode when the correct person find them again in the correct moment. $\endgroup$ – Hvjurthuk Aug 14 at 22:30
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    $\begingroup$ @Hvjurthuk The same paper by Ehresmann also introduced species, which arguably only became significant in the 80s with the work of André Joyal on combinatorial species. $\endgroup$ – Michael Greinecker Aug 14 at 22:37
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Julia sets were relatively obscure and little known until the advent of personal computing, when the ability to graph them in detail made it clear how amazing they are.

They now command a global audience of fans, even with little or no knowledge of mathematics.

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What about:

  • from Euler characteristic (special problem, thus "small") to homology theory of cw-complexes
  • from Galois (special problem, thus "small") to group theory and modern algebra?
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    $\begingroup$ The problem of solving polynomial equations by radicals was considered a central one, so I would not say that Galois theory began "small" $\endgroup$ – Francesco Polizzi Aug 14 at 21:30
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    $\begingroup$ I actually was going to comment the same as @FrancescoPolizzi. In a sense these problems were actually central to mathematics then and I would not in general consider work done by Gauss or Euler small in any sense as they are two of the brightest minds in mathematics. Anyway, thanks a lot for your answer! $\endgroup$ – Hvjurthuk Aug 14 at 22:18
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    $\begingroup$ @Hvjurthuk I do not think that Euler's theorem on polytopes was considered a central subject. $\endgroup$ – Fedor Petrov Aug 14 at 23:07
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    $\begingroup$ @FrancescoPolizzi I feel like Galois theory is an interesting case in that the more we learned about mathematics, the less important solving polynomials by radicals has seemed, but the more important Galois's insights have become (or at least they've stayed proportionally equally important by becoming relevant to more and more areas). Like it was lucky that people were initially so focused on solving a polynomial equation by radicals. $\endgroup$ – Will Sawin Aug 15 at 20:08
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    $\begingroup$ In fact, Galois theory (at least, in my opinion) is more an example of a mathematical discovery that killed a research field (solving polynomial equations over a field by radicals) or, better, radically transformed it in something much more deep and general. $\endgroup$ – Francesco Polizzi Aug 16 at 6:30
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Counting: one, two, three, four, ...

This originated and was put to practical use in prehistoric times.

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    $\begingroup$ Do you think that this discovery was "something very obscure and not looking very useful"? $\endgroup$ – Francesco Polizzi Aug 15 at 16:51
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    $\begingroup$ @FrancescoPolizzi : Perhaps not, but no one at the time could have anticipated the life of that idea. $\endgroup$ – Michael Hardy Aug 15 at 16:52
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    $\begingroup$ @MichaelHardy Okay, you went too far in (pre)history (even in biology itself). I am mostly interested in more modern things. However this answer is in the line of the famous quotation of Grothendieck about simple ideas. Legit though then. $\endgroup$ – Hvjurthuk Aug 15 at 17:23
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    $\begingroup$ Obligatory: smbc-comics.com/comic/a-new-method $\endgroup$ – Nate Eldredge Aug 16 at 19:50
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    $\begingroup$ @AmirSagiv, whether or not it's what @‍Hvjurthuk had in mind, this certainly doesn't seem like an attempt at a new question. $\endgroup$ – LSpice Aug 17 at 20:21
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In his book An Introduction to Combinatorial Analysis, Riordan observed that the number of ways to choose $k$ objects from $n$ objects, allowing repetition and disregarding order, can be written $(-1)^k{-n\choose k}$, while ${n\choose k}$ is the number of ways without repetition. This was the first inkling of the vast subject of combinatorial reciprocity. See for instance the book Combinatorial Reciprocity Theorems by Matthias Beck and Raman Sanyal.

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Introduction of $0$ in the place value system of counting. It was not given proper consideration in other places. So perhaps was pre-medieval mystery.

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I think the Durfee square is a good fit. The idea is incredibly simple - distinguish integer partitions based on the largest square that fits inside of them. The use of a square has since been extended to rectangles, staircases and likely some other shapes. The idea is indisputably a small one, but has proven itself an indispensable tool for working with integer partitions, hence also with hypergeometric functions and related topics. A quick search on Google Scholar shows many papers with Durfee square in the title, some of them quite recent. Note the top hit, which points out the Durfee square is identical to the h-index! One strike against the Durfee square for your question is that Durfee's advisor Sylvester recognized almost immediately how powerful a tool the Durfee square is in partition analysis. As he wrote in a letter to Cayley in 1883 (see Wikipedia),

"Durfee's square is a great invention of the importance of which its author has no conception."

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Two instances come to mind in digital signal processing (applied mathematics).

  1. The Fast Fourier Transform (FFT) computes the Discrete Fourier Transform in $O(N \log N)$ instead of $O(N^2)$. Supposedly, Gauss had a version of the FFT long before (electronic) computers made their impact.

  2. The second is the original wavelet transform, by A. Haar in 1909. Research in wavelet transforms has exploded since.

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I want to mention Selberg's integral, an $n$-dimensional generalization of Euler's beta integral. Selberg published it 1944 in Norwegian in the journal Norsk Matematisk Tidsskrift. Not surprisingly, it did not get a lot of publicity there. Later it was key to results in random matrix theory and other areas. There is an excellent article by Forrester and Warnaar summarizing the history and applications of the integral.

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Ramsey Theory has to be mentioned in this context I think. This is a somewhat obscure but interesting branch of combinatorics that is named after the mathematician/philosopher Frank Ramsey who proved its first result through Ramsey's theorem.

Interestingly, Ramsey only proved this theorem in passing as a minor lemma. He was actually trying to prove a decision problem for a particular model of first order logic, namely the Bernays–Schönfinkel class.

This lemma ended up spawning and entire sub-branch of mathematics and is mostly known for Ramsey Numbers, a class of numbers that are known to exist but are ridiculously hard to compute.

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Neural networks are a great example right now in machine learning. They were around for decades before the computing power to actually train them properly became available.

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    $\begingroup$ A similar candidate would be the Relu function $f(x) = max(0, x)$. Using this as an activation function instead of the sigmoid made it easier to train neural networks with more than 2 layers, which turn out to perform better in practice, for unknown reasons. This (together with computing power) has made all sorts of advances possible. There can't be many examples of such a simple mathematical object having such a massive impact on society. $\endgroup$ – Flounderer Sep 22 at 12:10
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Richard Stanely's 1973 paper "Linear homogeneous Diophantine equations and magic labelings of graphs" was the first time commutative algebra was used to study convex polytopes. But the paper is not really about polytopes per se. Rather, its main focus is on resolving the Anand-Dumir-Gupta conjecture about "magic squares," specifically, about the number $H_n(r)$ of $n\times n$ nonnegative integer matrices having all row and column sums equal to $r$. Part of the Anand-Dumir-Gupta conjecture was that for fixed $n$, the function $H_n(r)$ is a polynomial in $r$, which Stanley showed as a consequence of some basic theorems in commutative algebra going back to Hilbert.

As Stanley says in his personal account "How the Upper Bound Conjecture was Proved",

In this paper appears a geometric interpretation of MacMahon’s algorithm which, among other things, relates the polynomials $H_n(r)$ (and some more general polynomials) to certain triangulations of polytopes, in particular, the number $f_i$ of $i$-dimensional faces of such triangulations for all $i$. At that time I had no interest in the $f_i$’s themselves.

The situation changed not too long after that, however, when Stanley was able to use the commutative algebra connection to prove remarkable results in polytopal combinatorics, like the Upper Bound Conjecture for simplicial spheres.

Nowadays there is a whole subfield of combinatorial commutative algebra: commutative algebra is a basic tool in the study of polytopes (e.g., their face numbers); and conversely polytopal combinatorics provides commutative algebraists with interesting questions and examples.

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I think the implicit function theorem fits very well. The idea of solving an implicit equation is simple, an for examples like the circle one might call it a small idea. However, the implicit function theorem is still very useful and can be applied in various situation, for example to prove existence in complicated situations.

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If memory serves, in James Gleick's book Chaos, he describes the origins of this field as attempts to find numerical bugs and rounding errors in PDE solvers -- before it was realized that something far more profound was happening.

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Does Fermat's last theorem count? I mean, it was a small idea at first. For which he thought he had a proof, but it didn't fit in a small margin of paper. At the time, who would have thought this theorem will have such a deep impact on mathematics?

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    $\begingroup$ I wouldn't describe Fermat's last theorem as an idea; more of a problem. $\endgroup$ – Ben McKay Aug 17 at 11:06
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    $\begingroup$ @Ben it was a conjecture actually. Then became a theorem. I think it's fair to classify conjectures as ideas. $\endgroup$ – polfosol Aug 17 at 11:49
  • $\begingroup$ @Bumblebee what do you expect me to say? :) $\endgroup$ – polfosol Aug 18 at 6:02

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