Math Origins: The Totient Function | Mathematical Association of America

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Leonhard Euler's totient function, ϕ(n), is an important object in number theory, counting the number of positive integers less than or equal to n which are relatively prime to n. It has been applied to subjects as diverse as constructible polygons and Internet cryptography. The word totient itself isn't that mysterious: it comes from the Latin word tot, meaning "so many." In a way, it is the answer to the question "Quot?" ("how many"?). In The Words of Mathematics, Steven Schwartzman [Sch] notes that question/answer pairings like Quo/To are common linguistic constructions, similar to the Wh/Th pairing in English ("Where? There. What? That. When? Then."). Beyond that, there are two key questions: (1) how did the notation emerge and develop, and (2) what questions was it originally intended to answer?

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