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In mathematics, 1760 Plateau's problem were solved, but it was only in 1930 that general solutions were found in the context of mappings (immersions) independently by Jesse Douglas and Tibor Radó.

1760 - 1930 = 170 years.

What is the oldest open question solved in mathematics?

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The Delian problem (doubling the cube) and the angle trisection problem are probably the oldest ones, though their exact age is not known. (Same applies to all problems of constructions with ruler and compass). They were certainly around in 4 bc, and existing sources show that by that time they were already old. Since both problems were solved only in 19th century, we conclude that it took at least 22 centuries to solve them, probably longer. Of course there are also unsolved problems going back to the same times, like the problem about odd perfect numbers.

Some authors speculate that these problems are near the origin of mathematics, A. Seidenberg, The ritual origin of geometry".

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It's hard to say, but here are my two completely different suggestions.

  1. Is $\pi$ rational? The squaring of circle was already mentioned. But people (Egyptians and Babylonians) tried to calculate $\pi$ long before invention of ruler-and-compass constructions. Even if they didn't know the notion of irrational numbers, they probably would be interested in a problem whether a circumference of a unit circle could be described as "number" (e.g. sexagesimal number for Babylonians). So, you can argue that this problem is much older than squaring a circle. And, even if irrationality of $\pi$ was proved earlier (in 1760s) than transcendence of $\pi$ and impossibility of squaring a circle (in 1882), the irrationality of $\pi$ was probably much older open problem.

  2. But maybe the oldest open problem was a very simple one. For example, how to calculate the area of a given quadrilateral? Babylonians used an incorrect formula $\frac{a+c}2 \cdot \frac{b+d}2$, where $a$, $b$, $c$ and $d$ are the sides. I'm not sure when the correct formula was discovered, but it needs trigonometry, which was invented about 2nd century BC (depending on definition of invention of trigonometry). It's impossible to say when people first interested in calculating area of quadrilateral, but it is possible that it was more than two milleniums before asking a question about $\pi$. So it may be the oldest open question at the time when it was solved.

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  • $\begingroup$ +1 for considering problems that solved a long time ago. However, I suspect that the Babylonians would not consider the problem open - they just had a wrong solution and considered it solved? $\endgroup$ – Hagen von Eitzen Jan 12 at 20:36
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    $\begingroup$ It's hard to know what they thought. But it's more likely they knew that it is only approximate formula. They knew the correct formula for the area of trapezoid and used it in the same texts. They had to noticed that these formulas give the different answers. And the fact that they choose correct one for trapezoids indicates that the knew that the other one in not exact. $\endgroup$ – Alexei Kopylov Jan 13 at 20:18

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