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I'm looking for problems that due to modern developments in mathematics would nowadays be reduced to a rote computation or at least an exercise in a textbook, but that past mathematicians (even famous and great ones such as Gauss or Riemann) would've had a difficult time with.

Some examples that come to mind are group testing problems, which would be difficult to solve without a notion of error-correcting codes, and -- for even earlier mathematicians -- calculus questions such as calculating the area of some $n$-dimensional body.

The questions have to be understandable to older mathematicians and elementary in some sense. That is, past mathematicians should be able to appreciate them just as well as we can.

Tito Piezas III
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aellab
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    Solvability by radicals and Galois theory.. I think it's only in the late 1870's that Galois theory has been understood! – mich95 May 06 '15 at 12:46
  • Please clarify how long ago the "past mathematicians" you are talking about lived. What we often consider recent advancements might still be technically 100-150 years old (such as Galois Theory). Are you looking for things even more recent than that? – JMoravitz May 06 '15 at 12:53
  • @JMoravitz: I am most interested in say, the past 250 years. But I am hesitant to restrict the question to a given period of time. – aellab May 06 '15 at 12:58
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    Factoring 20-digit numbers. – Gerry Myerson May 06 '15 at 12:59
  • I think the easiest way to find such examples is to look for elementary problems that have been solved as corollaries of major proven theorems (think of the Poincare conjecture, Fermat's Last Theorem, etc.) -- these would've been quite difficult to solve without the recent proof of this theorem. – Newb May 06 '15 at 12:59
  • @GerryMyerson: That is a development in computers, not in mathematics. :P – aellab May 06 '15 at 13:01
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    Computers are mathematics. – Gerry Myerson May 06 '15 at 13:02
  • Does computing the area under a parabola count? – Mark Fantini May 06 '15 at 13:07
  • I wonder how long the formula for the number of ways to write a natural number as the sum of four squares has been known. I think that might qualify. – Tobias Kildetoft May 06 '15 at 13:19
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    I don't understand why the question was put on hold. Certainly it is broad, but each answer can be objective, short and self-contained (as seen below). How can I make it fit MSE's rules? – aellab May 06 '15 at 14:02
  • @GerryMyerson So, then, tell me how to factor 20-digit numbers. If you need a computer, please describe how to build one. – Akiva Weinberger May 06 '15 at 15:23
  • @columbus8myhw: Gerry's comment was shorthand for how, if you strip away the physical aspects, the fundamentals of computers is just maths (with a nod to Turing). See Turing machine for more details. (It need not even be a machine, could be a group of people manipulating that "tape".) – Tito Piezas III May 06 '15 at 15:46
  • @TitoPiezasIII Yeah, I know. But he said that there is a way to factor 20-digit numbers that the old mathematicians didn't know. The problem is that, that requires the construction of a Turing machine — and the building of a physical one, if we want it to do anything for us. So, I suppose it is a problem whose solution is known, but it's not "easy to solve today". (Buying a pre-built computer is cheating.) – Akiva Weinberger May 06 '15 at 15:54
  • @Michael: You can start by voting to re-open your own question. (I swear the way some people eagerly close questions that can be given the benefit of the doubt, you'd think there was a heavenly reward for it.) – Tito Piezas III May 06 '15 at 16:13
  • @TitoPiezasIII: Unfortunately, it looks like I need more rep to vote to reopen my own question... – aellab May 06 '15 at 20:46
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    "I'm looking for problems that due to modern developments in mathematics would nowadays be reduced to a rote computation or at least an exercise in a textbook, but that past mathematicians (even famous and great ones such as Gauss or Riemann) would've had a difficult time with." Past mathematicians would have had trouble factoring a 20-digit number. Nowadays, it's a rote computation. Why is there an argument here? – Gerry Myerson May 07 '15 at 00:58

4 Answers4

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That there exist transcendental numbers. This was first shown by Liouville, who proved that Liouville's number: $$\sum_{i=0}^\infty10^{-i!}$$ is transcendental.

The "modern" proof would be due to Cantor:

There are countably many algebraic numbers and uncountably many reals. Therefore there exists a transcendental number.

Proving that Liouville's number is transcendental isn't so hard, but compared to the above it seems quite torturous.

Oscar Cunningham
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    The problem of squaring the circle was of immense interest to the ancient mathematicians, which we now know to be impossible due to the transcendence of $\pi$. That is not to say that the proof $\pi$ is transcendental is easy, but it is understood. – JMoravitz May 06 '15 at 13:03
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I would say that computing the Fourier coefficients of a tamed function is a triviality today even at an engineering math 101 level.

Ph. Davis and R. Hersh tell the long and painful story of Fourier series. I quote from their book:

"Fourier didn't know Euler had already done this, so he did it over. And Fourier, like Bernoulli and Euler before him, overlooked the beautifully direct method of orthogonality [...]. Instead, he went through an incredible computation, that could serve as a classic example of physical insight leading to the right answer in spite of flagrantly wrong reasoning."

(Fifth Ch. "Fourier Analysis".)

zoli
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This sum-of-squares theorem of Fermat may qualify as an example:

An odd prime $p$ is expressible as the sum of squares $x^2+y^2$ if and only if $p\equiv 1 \text{ mod } 4$.

You can read this Wikipedia article (as of the most recent update to this answer) to see the difference in mental effort in the original proof by Euler, as opposed to a modern treatment using the fact that the Gaussian integers are a Euclidean domain.

A dual example: I think Brouwer would be astonished and pleased to know that the Brouwer fixed point theorem can now be proven for the simplex (and, with more effort, for convex polytopes) with absolutely no knowledge of topology; just some affine geometry and combinatorial intuition to prove Sperner's Lemma, and basic analysis to translate to the continuous setting.

It's still not an "easy" proof but it is an example of a classical problem that we now can solve with considerably less machinery, instead of the above example, whose ease of proof can be chalked up to more machinery.

Eric Stucky
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  • Is this really a modern treatment, though, or a modern simplification? The question being whether Dedekind's proof is "easier" and more natural from the perspective of modern theory, as opposed to just being a very clever simplification enabled by modern tools (but which would've required large mental effort to come up with). – aellab May 06 '15 at 13:44
  • @Michael: Yeah, you can see some of the earlier edits where I was a bit more bold in my claim :P . I feel justified in saying that it's not "just" a simplification: Dedekind's proof, even after laying out all the details, is substantively different than Euler's. But there are definitely details which expand the length (and difficulty) of Dedekind's proof considerably. – Eric Stucky May 06 '15 at 13:49
  • What's a good resource to learn about this "Euclidean domain" thing? – Akiva Weinberger May 06 '15 at 18:37
  • Any good intro abstract algebra textbook should explain basic ring theory. – Eric Stucky May 09 '15 at 05:47
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In the nineteenth century expressing the antiderivative of an elementary function as an elementary function was an open problem.

Nowadays, Risch algorithm, which can be run on machines, decides whether such operation can be done and, if so, yields a version of the correct result.

I cannot speak for past mathematicians, but I think this is a useful tool.

Added: @columbus8myhw made a very important technical remark in the comments, which is also explained in the last part of the wikipedia article.

Vladimir Sotirov
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    Technically, the Risch algorithm can't fully be run on a computer — because it involves deciding whether a given equation is always equal to $0$, which is an unsolvable problem. (IIRC.) But there are heuristic algorithms, so it can be run well enough. – Akiva Weinberger May 06 '15 at 15:07