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For any (a, b) satisfying the given condition, let k = a 2 + b 2 + 1 / ab and rearrange and substitute to get x 2 − (kb) x + (b 2 + 1) = 0. One root to this quadratic is a, so by Vieta's formulas the other root may be written as follows: x 2 = kb − a = b 2 + 1 / a . The first equation shows that x 2 is an integer and the ...
Viète's formula may be rewritten and understood as a limit expression [3] = =, where = = +.. For each choice of , the expression in the limit is a finite product, and as gets arbitrarily large, these finite products have values that approach the value of Viète's formula arbitrarily closely.
A method similar to Vieta's formula can be found in the work of the 12th century Arabic mathematician Sharaf al-Din al-Tusi. It is plausible that the algebraic advancements made by Arabic mathematicians such as al-Khayyam, al-Tusi, and al-Kashi influenced 16th-century algebraists, with Vieta being the most prominent among them. [2] [3]
Vieta may refer to: François Viète (1540–1603), commonly known by the Latin form of his name Franciscus Vieta, a French mathematician; Vieta (crater), a crater on the Moon, named after him; Vieta's formulas, expressing the coefficients of a polynomial as signed sums and products of its roots. Artūras Vieta (born 1961), Lithuanian sprint canoer
The characteristic polynomial of a square matrix is an example of application of Vieta's formulas. The roots of this polynomial are the eigenvalues of the matrix . When we substitute these eigenvalues into the elementary symmetric polynomials, we obtain – up to their sign – the coefficients of the characteristic polynomial, which are ...
François Viète (French: [fʁɑ̃swa vjɛt]; 1540 – 23 February 1603), known in Latin as Franciscus Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to his innovative use of letters as parameters in equations.
Since , the factors of 5 are addressed by noticing that since the residues of modulo 5 follow the cycle ,,, and those of follow the cycle ,,,, the residues of modulo 5 cycle through the sequence ,,,. Thus, 5 ∣ 149 n − 2 n {\displaystyle 5\mid 149^{n}-2^{n}} iff n = 4 k {\displaystyle n=4k} for some positive integer k {\displaystyle k} .
Chapter 1. The language of symmetry – an introduction to the mathematical concept of symmetry and its relation to geometric groups. Chapter 2. A delightful fiction – an introduction to complex numbers and mappings of the complex plane and the Riemann sphere. Chapter 3. Double spirals and Möbius maps – Möbius transformations and their ...