Let p be a prime≡l (mod 5). If 2 is a quintic nonresidue (mod p) then 2p-1/5 ≡ α (mod p) for some fifth root of unity α5 (≢ 1) (mod p). Emma Lehmer has given an explicit expression for a5 in terms of a particular solution of a certain quadratic partition of p. In this paper we show how in principle the corresponding result can be obtained for any quintic nonresidue D(mod p). Full details are given for D=2, 3, 5.

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Journal Pacific Journal of Mathematics
Williams, K.S. (1975). On euler’s criterion for quintic nonresidues. Pacific Journal of Mathematics, 61(2), 543–550.