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Criterion for 2 to be an \(\ell\)th power. (English) Zbl 0539.10006

Let \(\ell\) be an odd prime and let p be a prime \(\equiv 1\) (mod \(\ell)\). The authors prove that 2 is an \(\ell th\) power modulo p if and only if \(a_ 1+a_ 2+...+a_{\ell -1}\equiv 0\) (mod 2) where \((a_ 1,a_ 2,...,a_{\ell -1})\) is one of the exactly \(\ell -1\) solutions of the diophantine system of equations \[ (i)\quad p=\sum^{\ell - 1}_{i=1}a^ 2_ i-\sum^{\ell -1}_{i=1}a_ ia_{i+1},\quad \sum^{\ell -1}_{i=1}a_ ia_{i+1}=\sum^{\ell -1}_{i=1}a_ ia_{i+2}=...=\sum^{\ell -1}_{i=1}a_ ia_{i+\ell -1}, \] \(p\nmid \prod_{\lambda(2k)>k}(\sum^{\ell -1}_{i=1}a_ i\xi^ i)^{\sigma_ k}\), where \(\xi =e^{2\pi i/\ell}\) and where \(\lambda\) (n) is the least non-negative residue of n modulo \(\ell\) and \(\sigma_ k\) is the automorphism: \(\xi \mapsto \xi^ k\), \(1+a_ 1+...+a_{\ell - 1}\equiv 0\) (mod \(\ell)\), \(a_ 1+2a_ 2+...+(\ell -1)a_{\ell - 1}\equiv 0\) (mod \(\ell)\) where in (i) the subscripts of the a’s are taken modulo \(\ell\) and \(a_ 0=0\). This result extends the work of earlier authors for the cases \(\ell =3\) (Jacobi), \(\ell =5\) (Lehmer), \(\ell =7\) (Leonard and Williams), \(\ell =11\) (Leonard, Mortimer and Williams).
Reviewer: K.S.Williams

MSC:

11A15 Power residues, reciprocity
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