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2020
The Pythagorean theorem constitutes one of the first great ideas of mathematics. Its importance is evident in the fact that it has been taught in schools throughout human history; it has had many applications in science and engineering; it has cropped up in numerous other mathematical ideas; it has led to discoveries, such as Fermat’s last theorem; and
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The Pythagorean theorem constitutes one of the first great ideas of mathematics. Its importance is evident in the fact that it has been taught in schools throughout human history; it has had many applications in science and engineering; it has cropped up in numerous other mathematical ideas; it has led to discoveries, such as Fermat’s last theorem; and
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Reframing the Pythagorean Theorem
The College Mathematics Journal, 2019According to the great German astronomer and mathematician Johannes Kepler in his cosmological treatise Mysterium Cosmographicum [5] on the Copernican system, “Geometry has two great treasures: One...
Ian M. Adelstein, George L. Ashline
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2015
The Pythagorean Theorem is one of the oldest, best known, and most useful theorems in all of mathematics, and it has also surely been proved in more different ways than any other. Euclid gave two proofs of it in the Elements, as Proposition I,47, and also as Proposition VI,31, a more general but less well-known formulation concerning arbitrary ‘figures’
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The Pythagorean Theorem is one of the oldest, best known, and most useful theorems in all of mathematics, and it has also surely been proved in more different ways than any other. Euclid gave two proofs of it in the Elements, as Proposition I,47, and also as Proposition VI,31, a more general but less well-known formulation concerning arbitrary ‘figures’
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A Spherical Pythagorean Theorem
The Mathematical Intelligencer, 2010for a spherical right triangle with hypotenuse c and legs a and b, is generally presented as the ‘spherical Pythagorean theorem’. Still, it has to be remarked that this formula does not have an immediate meaning in terms of areas of simple geometrical figures, as the Pythagorean theorem does.
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An omega theorem on pythagorean triples
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1993Let \(A(x)\) denote the number of Pythagorean triples \((r,s,n)\) with \(r^ 2+ s^ 2= n^ 2\) and \(1\leq n\leq x\). Then \[ A(x)+ {\textstyle {4\over\pi}} x\log x+Bx+ E(x), \] where \(B\) is a well-defined constant. The remainder \(E(x)\) can be estimated by \(E(x)= O(\sqrt{x})\). Somewhat better results can be found in the papers of \textit{M.
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Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators
Nature Machine Intelligence, 2021Lu Lu, Pengzhan Jin, Guofei Pang
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Experimental quantum key distribution certified by Bell's theorem
Nature, 2022David Nadlinger +2 more
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