By Marvin Marcus, Henryk Minc

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As before, varying m on Q , we obtain all rational points on our cubic curve. In the elusive case where P has three distinct roots, the method of rational slopes does not work. A cubic rational curve in WeierP(x), where the cubic polynomial P has no double strass form y2 or triple roots, is called elliptic. (♦ The name “elliptic” comes from the fact that when trying to determine the circumference of an ellipse one encounters elliptic integrals of the form R(x, y)dx, 3. ) By the discussion above, an elliptic curve is everywhere nonsingular.

Remark. A much harder ancient problem is to ﬁnd a simple test to determine whether a given positive rational number r is a congruent number, the area of a right triangle with rational side lengths a, b, and c. In general, any nonzero rational number r can be made a square free integer by multiplying r by the square of another rational number s. As the name suggests, square free means that the integer (in this case s2 r) has no nontrivial square integral divisors. Now, if r is the area of a triangle with side lengths a, b, c, then s2 r is the area of a similar triangle with side lengths sa, sb, sc.

Generalize the ﬁrst equation to prove Lewis Caroll’s identity tn tn+s + tn+t iff st n2 + 1, and Machin’s formula t1 4t5 − t239 . The latter translates into π/4 11 4 tan−1 (1/5) − tan−1 (1/239). For more details, see J. Conway and R. Guy, The Book of Numbers, Springer, 1996. ) Derive the last formula using trigonometry. (Let tan θ 1/5 and use the double angle formula for the tangent to work out tan(2θ) 5/12 and tan(4θ) 120/119. Notice that this differs from tan(π/4) 1 by 1/119. ) 15. ♦ Given a basis {v, w} in R2 , the set L {kv + lw | k, l ∈ Z} is called a lattice in R2 with generators v and w.