By Paulo Ribenboim

In 1995, Andrew Wiles accomplished an explanation of Fermat's final Theorem. even if this was once definitely an excellent mathematical feat, one shouldn't brush aside prior makes an attempt made by way of mathematicians and smart amateurs to resolve the matter. during this ebook, geared toward amateurs interested in the heritage of the topic, the writer restricts his consciousness completely to simple equipment that experience produced wealthy effects.

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**Example text**

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.

Paris, 61 (1865), 921–924 and 961–965. , Sur la d´ecomposition d’un nombre entier en une somme de deux cubes rationnels, J. Math. , (2), 15 (1870), 217–236. , Mathematical Notes, Proc. Roy. Soc. Edinburgh, 7 (1872), 144. , Sur√certains nombres complexes compris dans la formule a + b −c, J. Math. Pures Appl. (3), 1 (1875), 317–372. , Uber die unbestimmte Gleichung x3 + y 3 = z 3 , Sitzungsber. B¨ohm Ges. , 1878, pp. 112–120. , Sur l’´equation ind´etermin´ee X 3 + Y 3 = AZ 3 , Nouv. Ann. , (2), 17 (1879), 425–426.