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Elliptic curves number theory

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Elliptic Curves & Number Theory R. Sujatha School of Mathematics TIFR 1 Aim: To explain the connection between a simple ancient problem in number theory and a deep sophisticated conjecture about Elliptic Curves (‘arithmetic Geometry’). 1 Aim: To explain the connection between a simple ancient problem in number theory and a deep sophisticated conjecture about Elliptic Curves (‘arithmetic Geometry’). Notation: N : set of natural numbers (1, 2, 3, . . .) 1 Aim: To explain the connection between a simple ancient problem in number theory and a deep sophisticated conjecture about Elliptic Curves (‘arithmetic Geometry’). Notation: N : set of natural numbers (1, 2, 3, . . .) Z : set of integers (. . . , −3, −2, −1, 0, 1, 2, . . .) 1 Aim: To explain the connection between a simple ancient problem in number theory and a deep sophisticated conjecture about Elliptic Curves (‘arithmetic Geometry’). Notation: N : set of natural numbers (1, 2, 3, . . .) Z : set of integers (. . . , −3, −2, −1, 0, 1, 2, . . .) Q : Rational numbers 2 • Recall that a real number is rational if it can be expressed in the form α = m n , where m and n are in Z. • Irrational numbers: Those which cannot be expressed in the form m/n, m, n ∈ Z. Example: √ √ 2, 1+ 5 π = 3.1419, . 2 3 Pythagorean Triples: • P QR is right angled triangle • Sides have lengths a, b, c. P b Q :: :: :: :: :: :: :: :: :: :: :: :: :: :: :: :: :: :: :: :: : c a R 4 Pytho ...
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