Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4595298 | Journal of Number Theory | 2008 | 41 Pages |
Abstract
Let be a prime. Let a,b∈Z with p∤a(a2+b2). In the paper we mainly determine by assuming p=c2+d2 or p=Ax2+2Bxy+Cy2 with AC−B2=a2+b2. As an application we obtain simple criteria for εD to be a quadratic residue , where D>1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field with negative norm. We also establish the congruences for and obtain a general criterion for p|U(p−1)/4, where {Un} is the Lucas sequence defined by U0=0, U1=1 and Un+1=bUn+k2Un−1 (n⩾1).
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory