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So Is it true that Quadratic residue was published and discovered before Legendre symbol and Euler's Criteria? Quadratic residue came in 1801 by Gauss(1). can you put these concepts in chronological order and by whom these concepts was first introduced or developed in paper form?

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I'd say yes. My reason being that the Legendre symbol simply defines if a number $\mod p$ is a quadratic residue to $p$ or not. There is no way to define a number being a quadratic residue $\mod p$ if you don't even know what a quadratic residue is.

Regarding the Euler criterion, that's a little bit more tricky. That too would've come after the discovering quadratic residues, but I'm not sure if that was before or after the Legendre symbol

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  • $\begingroup$ Just reminder: "...Euler and Legendre did not have Gauss's congruence notation, nor did Gauss have the Legendre symbol... " ([1]) [1]:en.wikipedia.org/wiki/… So this means that Legendre symbol and notation of Euler's criterion, where Legendre symbol is used must be after 1801 when Gauss brought this Quadratic_reciprocity - law. Or at least Gauss didn't know them yet in 1801. $\endgroup$ – user2723 Jun 3 '13 at 17:28
  • $\begingroup$ But Euler's criterion is already in 1748[wikipedia] in Euler's paper and Legendre presented his symbol in 1798[wikipedia]. But there were Fermat, Euler, Lagrange, Legendre, and other number theorists of the 17th and 18th centuries who proved some theorems and made some conjectures about quadratic residues, before first systematic treatment by Gauss in Disquisitiones Arithmeticae (1801). So in reality chronological order would be: Quadratic res., Euler's criterion( without Legendre symbol) and Legendre symbol. $\endgroup$ – user2723 Jun 3 '13 at 17:29

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