A quadratic partition of primes ≡1 (𝑚𝑜𝑑7)

Williams, Kenneth S.

doi:10.1090/S0025-5718-1974-0345902-2

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ISSN 1088-6842(online) ISSN 0025-5718(print)

A quadratic partition of primes $\equiv 1$ $({\rm mod}\ 7)$

Author: Kenneth S. Williams
Journal: Math. Comp. 28 (1974), 1133-1136
MSC: Primary 10A40
MathSciNet review: 0345902
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Abstract: The solutions of a quadratic partition of primes $p \equiv 1 \pmod 7$ , in terms of which the author and P. A. Leonard have given the cyclotomic numbers of order seven and also necessary and sufficient conditions for 2, 3, 5 and 7 to be seventh powers $\pmod p$ , are obtained for all such primes .

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[2] L. E. Dickson, Cyclotomy and trinomial congruences, Trans. Amer. Math. Soc. 37 (1935), no. 3, 363–380. MR 1501791, http://dx.doi.org/10.1090/S0002-9947-1935-1501791-3
[3] E. KUMMER, "Sur les nombres complexes qui sont formés avec les nombres entiers réels et les racines de l'unité," J. Analyse Math., v. 12, 1847, pp. 185-212.
[4] Philip A. Leonard and Kenneth S. Williams, The septic character of 2, 3, 5 and 7, Pacific J. Math. 52 (1974), 143–147. MR 0364064 (51 #319)
[5] P. A. LEONARD & K. S. WILLIAMS, "A diophantine system of Dickson," Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (To appear.)
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[7] K. G. REUSCHLE, "Zerfällung aller Primzahlen innerhalb des ersten Tausend in ihre aus siebenten Wurzeln der Einheit gebildeten complexen Primfactoren," Monatsh. Kl. Preuss. Akad. Wiss. Berlin, v. 1859, pp. 694-697.

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