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Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? …

Irrationality of $2^{1/n}$ for $n\geq 3$: if $2^{1/n}=p/q$ then $p^n = q^n+q^n$, contradicting Fermat's Last Theorem. Unfortunately FLT is not strong enough to prove $\sqrt{2}$ irrational. …

$ Many mathematician's gut reaction to the question is that the answer must be yes and Minkowski's uniqueness theorem provides some mathematical justification for such a belief---Minkwoski's uniqueness … theorem implies that an origin-symmetric star body in $\mathbb{R}^n$ is completely determined by the volumes of its central hyperplane sections, so these volumes of central hyperplane sections do contain …

But $$\int_{\bf R} \lim_{n \to \infty} f_n(x)\ dx = 0 \neq 1 = \lim_{n \to \infty} \int_{\bf R} f_n(x)\ dx,$$ contradicting the dominated convergence theorem. …

Theorems, questions, exercises, examples as well as definitions can be coming from an incorrect view of a subject! …

What he needed from GRH is the lack of real zeros in the interval $(53/54,1)$. m) Removing a condition in the Brauer-Siegel theorem. … That is, GRH implies the Brauer-Siegel theorem holds for sequences of number fields $K_n$ fitting condition (i). n) Lower bounds on root discriminants. …

Now let me just briefly summarize the main theorems on the basic nature of perfectoid spaces. … This leads to an improvement on Faltings's almost purity theorem: Theorem: Let $R$ be a perfectoid $K$-algebra, and let $S/R$ be finite étale. …

It is a counterexample to an 1899 "Theorem" of C.N. Little (Yale PhD, 1885), accepted as true by P.G. … The Perko pair have different writhes, and so Little's "Theorem", if true, would prove them to be distinct! …

It is a very complete book even introducing some needed commutative algebra and preparing the reader to learn arithmetic geometry like Mordell's conjecture, Faltings' or even Fermat-Wiles Theorem. … As a fundamental complement check Hauser's wonderful paper on the Hironaka theorem. …

Tarski's theorem on the non-definability of truth shows that there is no first-order definition that allows us a uniform treatment of saying that a particular particular formula $\varphi$ is true at a … In recent work (soon to be submitted for publication), Jonas Reitz, David Linetsky and I have proved the following theorem: Theorem. …

Seen on https://legauss.blogspot.com/2012/05/para-rir-ou-para-chorar-parte-13.html Theorem: $5!/2$ is even. Proof: $5!/2$ is the order of the group $A_5$. … But the Feit-Thompson Theorem asserts that every finite group with odd cardinal is solvable, so $5!/2$ must be an even number. …

They produce complex webs of definitions, theorems, and constructions, therefore $V$ should support management of large collections of formalized mathematics. … For instance, saying that a body of mathematics is "just a series of definitions, theorems and proofs" is a naive idealization that works in certain contexts, but certainly not in practical formalization …

Irving Segal proved in “Equivalences of measure spaces” (see also Kelley's “Decomposition and representation theorems in measure theory”) that for an enhanced measurable space $(S,M,N)$ that admits a faithful … The best argument for such restrictions is the following Gelfand-type duality theorem for commutative von Neumann algebras. Theorem. The following 5 categories are equivalent. …

See Theorem 1 in Cours D'Analyse Chap. VI Section 1. Five years later Abel pointed out that certain Fourier series are counterexamples. … See Graviton's answer for another assessment of Cauchy's work, which operated with continuity using infinitesimals in such a way that Abel's counterexample was not a counterexample to Cauchy's theorem. …

If mathematics were done by computers that mindlessly searched for theorems and proof but sometimes made mistakes in their proofs, then I expect that it would collapse. … For instance, while working on a recent project I discovered no fewer than nine mistaken theorem statements (not just mistakes in proofs of correct theorems) in published or almost-published literature …