By M. N. Huxley
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Extra resources for Distribution of Prime Numbers: Large Sieves and Zero-density Theorems
Special Cases (4) If the discriminant of a cubic polynomial with rational coefﬁcients is the sixth power of a√nonzero rational √ number, then 3 sin(2π/9), its roots are of the form r + s 3 sin(π/9), r + s √ r − s 3 sin(4π/9). Proof. (1) If x, y are integers and (x, y) satisﬁes 4x3 −3my 2 = m3 then letting u = −2x, v = y + m then u3 + v 3 = −8x3 + y 3 + 3y 2 m + 3ym2 + y 3 = −2m3 − 6my 2 + y 3 + 3y 2 m + 3ym2 + y 3 = (y − m)3 . Thus either x = 0 (which would imply −3y 2 = m2 , an absurdity) or y = ±m; in this case we have necessarily x = m.
For the equivalence of (1) and (2) we compute the Legendre symbol, using Gauss’ reciprocity law: −3 p = −1 p 3 p = (−1)(p−1)/2 (−1)(p−1)/2 p 3 = p . 3 So (−3/p) = +1 if and only if p ≡ 1 (mod 3), that is, p ≡ 1 (mod 6). For the equivalence of (2) and (3), we write X2 + X + 1 = X + 1 2 2 + 34 . 28 I. Special Cases 2 If there exists α ∈ Fp such that α2 + α + 1 = 0 then −3 = 4 α − 12 and conversely, if −3 = β 2 with β ∈ Fp , we take α = − 12 + β/2 so α2 + α + 1 = 0. 2. If k is a nonzero integer, if p is a prime, and p = c2 + 3d2 ∈ S, pk = a2 + 3b2 ∈ S then p divides ac ± 3bd and ad ∓ bc (with corresponding signs ) and k= ac ± 3bd p 2 +3 ad ∓ bc p ∈ S.
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