6 1. Introduction and, if n is even, #{(x, y, z) ∈ Z3 : n 2 = 4x2 + y2 + 32z2} = 1 2 #{(x, y, z) ∈ Z3 : n 2 = 4x2 + y2 + 8z2} . Moreover, if the Birch and Swinnerton-Dyer conjecture is true, then, conversely, these equalities imply that n is a congruent number. For example, for n = 2 we have n 2 = 1 = 4x2 + y2 + 32z2 if and only if x = z = 0 and y = ±1, so the left-hand side of the appropriate equation in Tunnell’s theorem is equal to 2. However, the right-hand side is equal to 1 and the equality does not hold. Hence, 2 is not a congruent number. For a complete historical overview of the congruent number prob- lem, see [Dic05], Ch. XVI. The book [Kob93] contains a thorough modern treatment of the problem. The reader may also find useful an expository paper [Con08] on the congruent number problem, written by Keith Conrad. Another neat exposition, more computational in nature (using Sage), appears in [Ste08], Section 6.5.3. Example 1.1.5 (Fermat’s last theorem). Let n ≥ 3. Are there any solutions to xn + yn = zn in integers x, y, z with xyz = 0? The answer is no. In 1637, Pierre de Fermat wrote in the margin of a book (Diophantus’ Arithmetica see Figure 9 in Section 5.5) that he had found a marvellous proof, but the margin was too small to contain it. Since then, many mathematicians tried in vain to demonstrate (or disprove!) this claim. A proof was finally found in 1995 by Andrew Wiles ([Wil95]). We shall discuss the proof in some more detail in Section 5.5. For now, we will outline the basic structure of the argument. First, it is easy to show that, to prove the theorem, it suﬃces to show the cases n = 4 and n = p ≥ 3, a prime. It is not diﬃcult to show that x4 +y4 = z4 has no non-trivial solutions in Z (this was first shown by Fermat). Now, suppose that p ≥ 3 and a, b, c are integers with abc = 0 and ap + bp = cp. Gerhard Frey conjectured that if such a triple of integers exists, then the elliptic curve E : y2 = x(x − ap)(x + bp)

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