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An Upper Bound on the Least Inert Prime in a Real Quadratic Field

  • Andrew Granville (a1), R. A. Mollin (a2) and H. C. Williams (a3)

Abstract

It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant $D\,>\,3705$ , there is always at least one prime $p\,<\,\sqrt{D}/2$ such that the Kronecker symbol $(D/P)\,=\,-1$ .

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References

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[1] Bach, E., Explicit bounds for primality testing and related problems. Math. Comp. 55(1990), 355380.
[2] Burgess, D. A., n character sums and L-series, I. Proc. LondonMath. Soc. 12(1962), 193206.
[3] Davenport, H., Multiplicative Number Theory. 2nd edn, Springer-Verlag, New York, 1980.
[4] Lukes, R. F., Patterson, C. D. and Williams, H. C., Some results on pseudosquares. Math. Comp. 65(1996), 361372.
[5] Mollin, R. A., Quadratics. CRC Press, Boca Raton, 1995.
[6] Norton, K. K., Bounds for sequences of consecutive power residues. Analytic Number Theory, Proc. Sympos. Pure Math. 24, Amer.Math. Soc., Providence, RI, 1973, 213220.
[7] Rosser, J. B. and Schoenfeld, L., Approximate formulae for some functions of prime numbers. Illinois J. Math. 6(1962), 6494.
[8] Schoenfeld, L., Sharper bounds for the Chebyshev functions θ(x) and ψ(x). Math. Comp. 30(1976), 337360. 900.
[9] Western, A. E. and Miller, J. C. P., Tables of Indices and Primitive Roots. Royal Society, Cambridge, 1968.
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