Zhi-Hong Sun
Published 2014-01-02Version 1
Let $\Bbb Z$ be the set of integers, and let $p$ be a prime of the form $4k+1$. Suppose $q\in\Bbb Z$, $2\nmid q$, $p\nmid q$, $p=c^2+d^2$, $c,d\in\Bbb Z$ and $c\equiv 1\pmod 4$. In this paper we continue to discuss congruences for $q^{[p/8]}\pmod p$ and present new reciprocity laws, but we assume $4p=x^2+qy^2$ or $p=x^2+2qy^2$, where $[\cdot]$ is the greatest integer function and $x,y\in\Bbb Z$.
Comments: 28 pages. arXiv admin note: substantial text overlap with arXiv:1108.3027
Congruences involving $\binom{4k}{2k}$ and $\binom{3k}k$
Congruences involving $\binom{2k}k^2\binom{3k}km^{-k}$
Congruences involving $\binom{2k}k^2\binom{4k}{2k}m^{-k}$
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