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In my opinion, this formula has value in teaching Mathematics (not only to kids). It can be used to illustrate what exactly is Math (as a science of rigorous problem statements and reasonnings). It can be used to explain that Math does not reduce to, or necessarily imply, complicated formulas. It can be used to explain the deep implications of the question “...


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The formula is derived in Willans, On Formulae for the nth Prime Number (1964) (Mathematical Gazette vol. 48, no. 366, pp. 413-415), who references Dickson's History of the Theory of Numbers, ch. XVIII "for references to other formulae of this nature". He does not quite mean explicit formulas for $p_n$, but rather explicit formulas for functions ...


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