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Criteria for Solvability of Certain Congruences

Published online by Cambridge University Press:  20 November 2018

Joseph B. Muskat
Joseph B. Muskat*
Affiliation:
University of Pittsburgh
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Lower case italics will denote rational integers, while lower case Greek letters will denote algebraic integers. The Law of Quadratic Reciprocity can be formulated as follows:

If p and a are distinct odd primes, then a is a quadratic residue (mod p) if and only if ( — 1)(p-1)/2p is a quadratic residue (mod q).

Let Xp denote the non-principal quadratic character (mod p) and let ζp be a primitive pth. root of unity. Then

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 343 - 352
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Ankeny, Nesmith C., Criterion for rth. power residuacity, Pacific J. Math., 10 (1960), 1115– 1124.Google Scholar
2. Bickmore, C. E., On the numerical factors of an — 1 (Second notice), Messenger Math., 26 (1896), 1–38.Google Scholar
3. Davenport, H. and H. Hasse, Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen, J. reine angew. Math., 172 (1935), 151–182.Google Scholar
4. Riesel, H., Note on the congruence ap-1 ≡ 1 (mod p2), Math, of Comp., 18 (1964), 149–150.Google Scholar
5. Muskat, J. B., Criteria for prime power residuacity, Doctoral Dissertation, M.I.T. (1961).Google Scholar
6. Muskat, J. B., On certain prime power congruences, Abh. Math. Sem. Univ. Hamburg, 26 (1963), 102–110.Google Scholar
7. Pepin, T., Mémoire sur les lois de réciprocité relatives aux residues de puissances, Pontif. accad. sci., Rome, 31 (1877), 40–148.Google Scholar
8. Lloyd Tanner, H. W., On the binomial equation xp — 1 = 0: quinquesection, Proc. Lond. Math. Soc, 18 (1887), 214–234.Google Scholar
9. Lloyd Tanner, H. W., On complex primes formed with the fifth roots of unity, Proc. Lond. Math. Soc, 24 (1893), 223–272.Google Scholar