From: Franz Lemmermeyer Subject: Re: Q:cubic residue Date: Thu, 14 Oct 1999 17:39:48 +0200 Newsgroups: sci.math.research mihai cipu wrote: > > Do you know handy characterizations of primes p such that 3 is a > cubic(or biquadratic, sextic,...) residue of p? http://www.rzuser.uni-heidelberg.de/~hb3/rec.html http://www.rzuser.uni-heidelberg.de/~hb3/recbib.html should be helpful. franz ============================================================================== From: Bob Silverman Subject: Re: Q:cubic residue Date: Mon, 18 Oct 1999 19:00:38 GMT Newsgroups: sci.math.research In article <3805F944.25E0@mpim-bonn.mpg.de>, Franz Lemmermeyer wrote: [previous article quoted --djr] It is also an elementary observation that every integer is a cubic residue of primes congruent to -1 mod 6. Bob Silverman "You can lead a horse's ass to knowledge, but you can't make him think" Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: Robin Chapman Subject: Re: Q:cubic residue Date: Tue, 19 Oct 1999 12:39:45 GMT Newsgroups: sci.math.research In article , mcipu@stoilow.imar.ro (mihai cipu) wrote: > Do you know handy characterizations of primes p such that 3 is a > cubic(or biquadratic, sextic,...) residue of p? > > Thank you for your kind help! Various criteria for n-ic residuacity of various small numbers including 3 in terms of divisibility properties of Jacobi sums can be found in Berndt, Hardy & Williams, _Gauss and Jacobi Sums_, Wiley-Interscience, 1998 http://math.ucsd.edu/~revans/errata/errata.html -- Robin Chapman http://www.maths.ex.ac.uk/~rjc/rjc.html "`Well, I'd already done a PhD in X-Files Theory at UCLA, ...'" Greg Egan, _Teranesia_ Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: lemmerm@mpim-bonn.mpg.de (Franz Lemmermeyer) Subject: Re: Octic residue character of 2. Date: 27 Sep 99 15:53:13 GMT Newsgroups: sci.math.numberthy > 2 is a biquadratic residue mod p if and only if there are integers > x,y such that x^2 + 64 y^2 = p. > > Are there similar (simple) results for the octic character, 2^4-ic > character, ... ? Similar yes, simple no. A. L. Whiteman, The sixteenth power residue character of 2, Can. J. Math. 6 (1954), 364--373; Zbl 55.27102 A. Aigner, Kriterien zum 8. und 16. Potenzcharakter der Reste 2 und -2, Deutsche Math. 4 (1939), 44--52; FdM 65 - I (1939), 112; H. Hasse, Der 2^n-te Potenzcharakter von 2 im Koerper der 2^n-ten Einheitswurzeln, Rend. Circ. Matem. Palermo (2), 7 (1958), 185--243 More references can be found on http://www.rzuser.uni-heidelberg.de/~hb3/recbib.html franz