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Congruences for \(q^{[p/8]}\pmod p\). (English) Zbl 1292.11007

Summary: Let \(\mathbb {Z}\) be the set of integers, and let \((m,n)\) be the greatest common divisor of the integers \(m\) and \(n\). Let \(p\equiv 1\pmod 4\) be a prime, \(q\in \mathbb {Z}\), \(2\nmid q\) and \(p=c^2+d^2=x^2+qy^2\) with \(c,d,x,y\in \mathbb {Z}\) and \(c\equiv 1\pmod 4\). Suppose that \((c,x+d)=1\) or \((d,x+c)\) is a power of 2. In this paper, by using the quartic reciprocity law, we determine \(q^{[p/8]}\pmod p\) in terms of \(c,d,x\) and \(y\), where \([\,\cdot\, ]\) is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers [J. Number Theory 129, No. 3, 499–550 (2009; Zbl 1183.11001), Acta Arith. 149, No. 3, 275–296 (2011; Zbl 1238.11005)].

MSC:

11A15 Power residues, reciprocity
11A07 Congruences; primitive roots; residue systems
11E25 Sums of squares and representations by other particular quadratic forms
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