On the Representation of Integers by Binary Cubic Forms of Positive Discriminant.
J.-H. Evertse (1983)
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J.-H. Evertse (1983)
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On the assumption of a Riemann hypothesis for certain Hasse-Weil L-functions, it is shewn that a quaternary cubic form f(x) with rational integral coefficients and non-vanishing discriminant represents through integral vectors x almost all integers N having the (necessary) property that the equation f(x)=N is soluble in every p-adic field ℚₚ. The corresponding proposition for quinary forms is established unconditionally.
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Christopher Hooley (1967)
Journal für die reine und angewandte Mathematik
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Attila Bérczes, Jan-Hendrik Evertse, Kálm'an Győry (2004)
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Jun Ho Lee, Stéphane R. Louboutin (2014)
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Let ϵ be a totally real cubic algebraic unit. Assume that the cubic number field ℚ(ϵ) is Galois. Let ϵ, ϵ' and ϵ'' be the three real conjugates of ϵ. We tackle the problem of whether {ϵ,ϵ'} is a system of fundamental units of the cubic order ℤ[ϵ,ϵ',ϵ'']. Given two units of a totally real cubic order, we explain how one can prove that they form a system of fundamental units of this order. Several explicit families of totally real cubic orders defined by parametrized families of cubic...