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Are there theorems earlier to the works of Abel and Galois showing that a problem that mathematicians had been trying to solve for a long time was actually unsolvable? Or are those works novel in the sense of establishing the inexistence, rather than the existence, of some highly desired entity?

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    $\begingroup$ Irrationality of the $\sqrt{2}$. Goes back to Pythagoreans, although they phrased it differently. The "highly desired entity" were two integers that are to each other as the side of a square is to its diagonal. Infinitude of primes. Probably also goes back to Pythagoreans, a proof appears in Elements. The "entity" was the largest prime. Archimedes, Apollonius, etc., routinely derived "symptoms" in construction problems, violation of which implied that the problem has no solution. $\endgroup$
    – Conifold
    Jun 30 at 11:19
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    $\begingroup$ You may be interested in Crippa's dissertation Impossibility results: from geometry to analysis, which is a comprehensive survey of early history, from antiquity to 17th century. $\endgroup$
    – Conifold
    Jul 1 at 6:38
  • $\begingroup$ @Conifold was a largest prime something "deeply valued" by Pythagoreans? If so, why? $\endgroup$ Jul 1 at 11:48
  • $\begingroup$ @Conifold The dissertation looks very promising! Thanks a lot for pointing it out. Would you please move your comment to an answer so it becomes broadly visible? $\endgroup$ Jul 1 at 12:37
  • $\begingroup$ @MichaelBächtold I do not think we know. Euclid thought it valued enough to include into Elements without ever using it later. $\endgroup$
    – Conifold
    Jul 1 at 14:11

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Some of the ancient examples.

Irrationality of $\sqrt{2}$. Goes back to Pythagoreans, although they phrased it differently. The "highly desired entity" were two numbers (positive integers) that are to each other as the side of a square to its diagonal. This would have supported the "all is number" thesis that Pythagoreans held, if we believe Aristotle (there are reasonable doubts, see Zhmud, All Is Number?). Stories about the "revolution" the negative result caused are likely exaggerated, at best, see Fowler, Mathematics of Plato's Academy.

Infinitude of primes. The "entity" was the largest prime. Probably also goes back to Pythagoreans, although many authors assert that Euclid came up with it on his own. There is no evidence either way. But he thought it important enough to include into Elements even though the proposition is not used anywhere later.

Archimedes, Apollonius, etc., routinely derived "symptoms" in construction problems, violation of which implied that the problem has no solution. For example, in On Sphere and Cylinder Archimedes considered the problem of cutting a sphere by a plane to divide its volume in a given ratio, reduced it to a cubic equation (in modern algebraic translation, he equated ratios), and derived conditions for those to have a solution.

Many more examples are discussed in Crippa's dissertation Impossibility results: from geometry to analysis, which is a comprehensive survey of early history, from antiquity to 17th century. He distinguishes between "absolute" impossibilities, such as the ones above, and impossibilities conditional on the tools employed, such as impossibility of duplicating a cube, trisecting an angle or squaring a circle with ruler and compass.

I am not sure the distinction is very crisp. The impossibility of squaring a circle followed by proving transcendentality of $\pi$, which is harder, but not conceptually different from proving irrationality of $\sqrt{2}$. But, in any case, those are closer to what Abel and Galois did, and were not proved until 19th century despite widespread belief since antiquity, and (flawed) earlier attempts starting with Descartes and Gregory, see Lutzen, The Algebra of Geometric Impossibility.

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    $\begingroup$ Long before irrationality of $\sqrt{2}$, there was impossibility of dividing numbers in the sexagesimal system. Babylonians did not use fractions of the form $\frac a b$. They use the sexagesimal system. But division in such system is not always possible. We don't know what Babylonians thought and taught about this. But they must know this fact, as they have table for inverse numbers. For number 59 they have 1/59=0; 1, 1, 1 .... So they should know that it is infinite. The fact that you can't write 1/59 should be as surprising for them as irrationality of $\sqrt{2}$ for Greeks. $\endgroup$ Jul 2 at 0:30
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    $\begingroup$ Is there really any record that anyone actually thought there are finitely prime numbers as a “deeply held” belief? $\endgroup$
    – KCd
    Jul 3 at 5:44

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