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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
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