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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]
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.
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 ...
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.
Vieta's substitution is a method introduced by François Viète (Vieta is his Latin name) in a text published posthumously in 1615, which provides directly the second formula of § Cardano's method, and avoids the problem of computing two different cube roots.
Richard Rusczyk (/ ˈ r ʌ s ɪ k /; Polish: [ˈrustʂɨk]; born September 21, 1971) is the founder and chief executive officer of Art of Problem Solving Inc. (as well as the website, which serves as a mathematics forum and place to hold online classes) and a co-author of the Art of Problem Solving textbooks.
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} .
I think the last sentence summarizes it well. In English at least, the tradition has been to use Vieta, and Viete is an overcorrection (outside historical or biographic contexts), like saying "Hero's formula" with Greek pronunciation, instead than Heron's formula. 73.89.25.252 04:52, 13 June 2020 (UTC)