## 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

### Keywords:

power residues; Jacobi sums
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