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A few remarks on cubic residues. (De residuis cubicis commentatio numerosa.) (Latin) ERAM 002.0047cj
Jacobi remarks that Gauß has announced a memoir on biquadratic residues, in which he will prove a criterion for $$2$$ to be a biquadratic residue of primes $$p$$. He observes that for cubic residuacity of primes $$p = 3n+1$$ one has to consider the representations $$4p = L^2 + 27M^2$$. He announces the following theorem: If $$p$$ and $$q$$ are prime numbers of the form $$3n+1$$, with $$4p = L^2 + 27M^2$$, and if $$x$$ is an integer with $$x^2 + 3 \equiv 0 \pmod q$$, then $$q$$ will be a cubic residue with respect to $$p$$ if and only if $$\frac{L+3mx}{L-3Mx}$$ is a cubic residue with respect to $$p$$.
His second theorem deals with primes $$q = 6n-1$$; he calls integers $$x$$ with $$x^{\frac{q+1}3} \equiv 1 \pmod p$$ cubic residues of $$q$$, and announces the following result: If $$p$$ is a prime of the form $$6n+1$$, if $$4p = L^2 + 27M^2$$, and if $$q$$ is a prime of the form $$6n-1$$, then $$q$$ will be a cubic residue with respect to $$p$$ if and only if $$\frac{L+3m\sqrt{-3}}{L-3M\sqrt{-3}}$$ is a cubic residue with respect to $$p$$.
In addition he remarks that if $$p = 3n+1$$ satisfies $$4p = L^2 + 27M^2$$, then $$L$$ is the minimal remainder modulo $$p$$ of $$- \frac{(n+1)(n+2)\cdots 2n}{1 \cdot 2 \cdots n} = - \binom{2n}{n}$$ that has the form $$3k+1$$, and gives a similar result for primes $$p = 7n+1$$.
Jacobi presented the proofs of these results in his lectures; see [F. Lemmermeyer (ed.) and H. Pieper (ed.), Vorlesungen über Zahlentheorie. Carl Gustav Jacob Jacobi, Wintersemester 1836/37, Königsberg. Augsburg: ERV Dr. Erwin Rauner Verlag (2007; Zbl 1148.11003)].

##### MSC:
 11A15 Power residues, reciprocity
##### Keywords:
cubic residues; cubic reciprocity law
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