A Pythagorean quadruple is a tuple of integers a, b, c, and d, such that a2 + b2 + c2 = d2. They are solutions of a Diophantine equation and often only...

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known...

the hypotenuse Pythagorean triple – a set of three positive integers that can occur in the Pythagorean theorem Pythagorean quadruple - a set of four...

square of the inverse of the integer altitude from the hypotenuse. Pythagorean quadruples are sets of four integers such that the sum of the squares of the...

equation attributed to Jacobi and Madden Prouhet–Tarry–Escott problem Pythagorean quadruple – Four integers where the sum of the squares of three equals the...

the components of the velocity vector are exactly those from the Pythagorean quadruple parametrization. Other choices of density and pressure are possible...

Jacobi–Madden equation Prouhet–Tarry–Escott problem Beal's conjecture Pythagorean quadruple Generalized taxicab number Sums of powers, a list of related conjectures...

\right\|\left\|\mathbf {B} \right\|}}\quad (-\pi <\theta \leq \pi )} The Pythagorean trigonometric identity then provides: ‖ A × B ‖ 2 + ( A ⋅ B ) 2 = ‖ A...

Jacobi–Madden equation Prouhet–Tarry–Escott problem Taxicab number Pythagorean quadruple Sums of powers, a list of related conjectures and theorems Distributed...

congruent number elliptic curve of rank at least 2. Pythagorean quadruple Wacław Sierpiński, Pythagorean Triangles, Dover Publications, 2003 (orig. ed. 1962)...

the 100 cents lower used in 12 TET, and the 113.7 cents lower used in Pythagorean tuning. In well temperaments, a flat is two or more different sizes,...

especially smallest solution for k = 4) Prouhet–Tarry–Escott problem Pythagorean quadruple Sums of powers, a list of related conjectures and theorems L. J...

of a semitone, an interval known as the Pythagorean comma. If limited to twelve pitches per octave, Pythagorean tuning markedly shortens the width of one...

equation Lander, Parkin, and Selfridge conjecture Taxicab number Pythagorean quadruple Sums of powers, a list of related conjectures and theorems Discrete...

Beal's conjecture Prouhet–Tarry–Escott problem Taxicab number Pythagorean quadruple Lander, Parkin, and Selfridge conjecture Sums of powers, a list...

the tempered sharp and flat. They presuppose a Pythagorean division of the octave taking the Pythagorean comma (about an eighth of the tempered tone, actually...

^{2}\left(\cos ^{2}\varphi +\sin ^{2}\varphi \right)=\rho ^{2}} using the Pythagorean trigonometric identity (can be useful to simplify this operation). The...

comma ≈ ⁠1/ 344 ⁠ pythagorean comma). Thus, 53 tone equal temperament is for all practical purposes equivalent to an extended Pythagorean tuning. After Mercator...

unsolved problems in mathematics) Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties. A general formula by...

serve beer in mustache cups." Fuddling cup Noggin (cup) Plastic cup Pythagorean cup Frost, Natasha (May 31, 2017). "A Look Back at the Mustache Cups...

just fifth is the basis of Pythagorean tuning; the difference between twelve just fifths and seven octaves is the Pythagorean comma. The sum of all n-choose...

Haidao Suanjing which dealt with using Pythagorean theorem (already known by the 9 chapters), and triple, quadruple triangulation for surveying; his accomplishment...

root quadruple of the largest four largest circles and four smallest curvatures. This quadruple can be found, starting from any other quadruple from the...

Sloane, N. J. A. (ed.). "Sequence A088959 (Lowest numbers which are d-Pythagorean decomposable, i.e., square is expressible as sum of two positive squares...

mathematical analysis. Mathematicians of Ancient Greece, according to the Pythagorean doctrine, understood calculation of area as the process of constructing...

Hanoi at cut-the-knot Real-time GPU generated Sierpinski Triangle in 3D Pythagorean triangles, Waclaw Sierpinski, Courier Corporation, 2003 A067771    Number...

made such rapid progress that he discovered an original proof of the Pythagorean theorem before his thirteenth birthday. A family tutor, Max Talmud, said...

between one musical note and another with one-quarter the wavelength or quadruple the frequency. It has also been referred to as the bisdiapason. The fourth...

{\displaystyle x} -axis of a semicircle (the square root is a consequence of the Pythagorean theorem), and the integral computes the area below the semicircle. The...

powers of 4 (45 + 40); Jacobsthal-Lucas number; hypotenuse of primitive Pythagorean triangle 1026 = sum of two distinct powers of 2 (1024 + 2) 1027 = sum...