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Quartic, octic residues and binary quadratic forms | | Zhi-Hong Sun
; | | Date: |
15 Aug 2011 | | Abstract: | Let $pequiv 1mod 4$ be a prime, $q$ be an odd number and
$p=c^2+d^2=x^2+qy^2$ for some integers $c,d,x$ and $y$. Suppose that $cequiv
1mod 4$ and $c$ is coprime to $x+d$. In the paper, by using the quartic
reciprocity law we determine $q^{[p/8]}mod p$ in terms of $c,d,x$ and $y$,
where $[cdot]$ is the greatest integer function. When $q=b^2+4^{alpha}$, we
also determine $ig(frac{b+sqrt{b^2+4^{alpha}}}2ig)^{frac{p-1}4}mod p$.
As an application we obtain the congruence for $U_{frac{p-1}4}mod p$ and the
criterion for $pmid U_{frac{p-1}8}$ (if $pequiv 1mod 8$), where ${U_n}$
is the Lucas sequence given by $U_0=0, U_1=1$ and
$U_{n+1}=bU_n+4^{alpha-1}U_{n-1} (nge 1)$. Hence we partially solve some
conjectures posed by the author in previous two papers. | | Source: | arXiv, 1108.3027 | | Services: | Forum | Review | PDF | Favorites | |
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