• The Mathematical Education
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• v.41 no.2
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• pp.227-231
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• 2002

The Pythagorean theorem and Pythagorean triple are well known. We know some Pythagorean triples, however we don't Cow well that every natural number can belong to some Pythagorean triple. In this paper, we show that every natural number, which is not less than 2, can be a length of a leg(a side opposite the acute angle in a right triangle) in some right triangle, and list some Pythagorean triples.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1133-1138
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• 2015

In 1956, $Je{\acute{s}}manowicz$ conjectured that, for any positive integer n and any primitive Pythagorean triple (a, b, c) with $a^2+b^2=c^2$, the equation $(an)^x+(bn)^y=(cn)^z$ has the unique solution (x, y, z) = (2, 2, 2). In this paper, under some conditions, we prove the conjecture for the primitive Pythagorean triples $(a,\;b,\;c)=(4k^2-1,\;4k,\;4k^2+1)$.

• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.251-274
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• 2020

We study a dynamical system that was originally defined by Romik in 2008 using an old theorem of Berggren concerning Pythagorean triples. Romik's system is closely related to the Farey map on the unit interval which generates an additive continued fraction algorithm. We explore some number theoretical properties of the Romik system. In particular, we prove an analogue of Lagrange's theorem in the case of the Romik system on the unit quarter circle, which states that a point possesses an eventually periodic digit expansion if and only if the point is defined over a real quadratic extension field of rationals.

• Journal for History of Mathematics
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• v.16 no.1
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• pp.17-24
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• 2003

The Pythagorean equation $x^2{＋}y^2{=}z^2$ and Pythagorean triple had appeared in the Babylonian clay tablet made between 1900 and 1600 B. C. Another quadratic equation called Pell equation was implicit in an Archimedes' letter to Eratosthenes, so called ‘cattle problem’. Though elliptic equation were contained in Diophantos’ Arithmetica, a substantial progress for the solution of cubic equations was made by Bachet only in 1621 when he found infinitely many rational solutions of the equation $y^2{=}x^3{-}2$. The equation $y^2{=}x^3{+}c$ is the simplest of all elliptic equations, even of all Diophantine equations degree greater than 2. It is due to Bachet, Dirichlet, Lebesque and Mordell that the equation in better understood.

• Bulletin of the Korean Mathematical Society
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• v.49 no.1
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• pp.175-195
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• 2012

Representing planar Pythagorean hodograph (PH) curves by the complex roots of their hodographs, we standardize Farouki's double cubic method to become the undetermined junction point (UJP) method, and then prove the generic existence of solutions for general $C^1$ Hermite interpolation problems. We also extend the UJP method to solve $C^2$ Hermite interpolation problems with multiple PH cubics, and also prove the generic existence of solutions which consist of triple PH cubics with $C^1$ junction points. Further generalizing the UJP method, we go on to solve $C^2$ Hermite interpolation problems using two PH quintics with a $C^1$ junction point, and we also show the possibility of applying the modi e UJP method to $G^2[C^1]$ Hermite interpolation.

• East Asian mathematical journal
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• v.28 no.4
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• pp.419-433
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• 2012

In this paper, by using the inductive method, recurrence relation, the unit circle, circle to inscribe a right-angled triangle, formula of multiple angles, solution of quadratic equation and Fibonacci numbers, we study various problem solving methods to find pythagorean triple.

• Journal for History of Mathematics
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• v.30 no.4
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• pp.221-232
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• 2017

In this paper, we discuss how ancient Egyptians find out the area of the circle based on $\ll$Ahmose Papyrus$\gg$. Vogel and Engels studied the quadrature of the circle, one of the basic concepts of ancient Egyptian mathematics. We look closely at the interpretation based on the approximate right triangle of Robins and Shute. As circumstantial evidence for Robbins and Shute's hypothesis, Egyptians prior to the 12th dynasty considered the perception of a right triangle as examples of 'simultaneous equation', 'unit of length', 'unit of slope', 'Egyptian triple', and 'right triangles transfer to Greece'. Finally, we present a method to utilize the squaring the circle by ancient Egyptians interpreted by Robbins and Shute as the dynamic symmetry of Hambidge.