Diophantine equations Henri Darmon McGill University CRM-ISM - PowerPoint PPT Presentation

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT Diophantine equations Henri Darmon McGill University CRM-ISM Colloquium, UQAM, January 8, 2010

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT Diophantine equations 1 Cubic equations 2 FLT 3 Pell’s equation 4 Elliptic curves 5 Back to FLT 6

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT Definition A Diophantine equation is a system of polynomial equations with integer coefficients: f 1 ( x 1 , . . . , x m ) = · · · = f n ( x 1 , . . . , x m ) = 0 , in which one is solely interested in the integer solutions. Some examples : 1 Cubic equations, like y 2 = x 3 + 1; 2 The Fermat-Pell equation: x 2 − Dy 2 = 1; 3 Fermat’s equation: x n + y n = z n ; 4 x 19 y 3 − 198 z 713 + 15 xyz 3 = 1098 w 2001 . A large part of number theory is concerned with the study of Diophantine equations.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT Definition A Diophantine equation is a system of polynomial equations with integer coefficients: f 1 ( x 1 , . . . , x m ) = · · · = f n ( x 1 , . . . , x m ) = 0 , in which one is solely interested in the integer solutions. Some examples : 1 Cubic equations, like y 2 = x 3 + 1; 2 The Fermat-Pell equation: x 2 − Dy 2 = 1; 3 Fermat’s equation: x n + y n = z n ; 4 x 19 y 3 − 198 z 713 + 15 xyz 3 = 1098 w 2001 . A large part of number theory is concerned with the study of Diophantine equations.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT Definition A Diophantine equation is a system of polynomial equations with integer coefficients: f 1 ( x 1 , . . . , x m ) = · · · = f n ( x 1 , . . . , x m ) = 0 , in which one is solely interested in the integer solutions. Some examples : 1 Cubic equations, like y 2 = x 3 + 1; 2 The Fermat-Pell equation: x 2 − Dy 2 = 1; 3 Fermat’s equation: x n + y n = z n ; 4 x 19 y 3 − 198 z 713 + 15 xyz 3 = 1098 w 2001 . A large part of number theory is concerned with the study of Diophantine equations.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT Definition A Diophantine equation is a system of polynomial equations with integer coefficients: f 1 ( x 1 , . . . , x m ) = · · · = f n ( x 1 , . . . , x m ) = 0 , in which one is solely interested in the integer solutions. Some examples : 1 Cubic equations, like y 2 = x 3 + 1; 2 The Fermat-Pell equation: x 2 − Dy 2 = 1; 3 Fermat’s equation: x n + y n = z n ; 4 x 19 y 3 − 198 z 713 + 15 xyz 3 = 1098 w 2001 . A large part of number theory is concerned with the study of Diophantine equations.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT Definition A Diophantine equation is a system of polynomial equations with integer coefficients: f 1 ( x 1 , . . . , x m ) = · · · = f n ( x 1 , . . . , x m ) = 0 , in which one is solely interested in the integer solutions. Some examples : 1 Cubic equations, like y 2 = x 3 + 1; 2 The Fermat-Pell equation: x 2 − Dy 2 = 1; 3 Fermat’s equation: x n + y n = z n ; 4 x 19 y 3 − 198 z 713 + 15 xyz 3 = 1098 w 2001 . A large part of number theory is concerned with the study of Diophantine equations.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT Definition A Diophantine equation is a system of polynomial equations with integer coefficients: f 1 ( x 1 , . . . , x m ) = · · · = f n ( x 1 , . . . , x m ) = 0 , in which one is solely interested in the integer solutions. Some examples : 1 Cubic equations, like y 2 = x 3 + 1; 2 The Fermat-Pell equation: x 2 − Dy 2 = 1; 3 Fermat’s equation: x n + y n = z n ; 4 x 19 y 3 − 198 z 713 + 15 xyz 3 = 1098 w 2001 . A large part of number theory is concerned with the study of Diophantine equations.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT Definition A Diophantine equation is a system of polynomial equations with integer coefficients: f 1 ( x 1 , . . . , x m ) = · · · = f n ( x 1 , . . . , x m ) = 0 , in which one is solely interested in the integer solutions. Some examples : 1 Cubic equations, like y 2 = x 3 + 1; 2 The Fermat-Pell equation: x 2 − Dy 2 = 1; 3 Fermat’s equation: x n + y n = z n ; 4 x 19 y 3 − 198 z 713 + 15 xyz 3 = 1098 w 2001 . A large part of number theory is concerned with the study of Diophantine equations.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT Some questions a number theorist might be asked This is the 21st century. Are there still questions about whole numbers that we don’t know how to answer? Isn’t the study of Diophantine equations just a recreational pursuit? Claim : Diophantine equations lie beyond the realm of recreational mathematics, because their study draws on a rich panoply of mathematical ideas. These ideas, and the new questions they lead to, are just as interesting (perhaps more!) than the equations which might have led to their discovery.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT Some questions a number theorist might be asked This is the 21st century. Are there still questions about whole numbers that we don’t know how to answer? Isn’t the study of Diophantine equations just a recreational pursuit? Claim : Diophantine equations lie beyond the realm of recreational mathematics, because their study draws on a rich panoply of mathematical ideas. These ideas, and the new questions they lead to, are just as interesting (perhaps more!) than the equations which might have led to their discovery.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT Some questions a number theorist might be asked This is the 21st century. Are there still questions about whole numbers that we don’t know how to answer? Isn’t the study of Diophantine equations just a recreational pursuit? Claim : Diophantine equations lie beyond the realm of recreational mathematics, because their study draws on a rich panoply of mathematical ideas. These ideas, and the new questions they lead to, are just as interesting (perhaps more!) than the equations which might have led to their discovery.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT Some questions a number theorist might be asked This is the 21st century. Are there still questions about whole numbers that we don’t know how to answer? Isn’t the study of Diophantine equations just a recreational pursuit? Claim : Diophantine equations lie beyond the realm of recreational mathematics, because their study draws on a rich panoply of mathematical ideas. These ideas, and the new questions they lead to, are just as interesting (perhaps more!) than the equations which might have led to their discovery.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT Diophantine equations 1 Cubic equations 2 FLT 3 Pell’s equation 4 Elliptic curves 5 Back to FLT 6

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT First example: the equation y 2 + y = x 3 Theorem The equation y 2 + y = x 3 has only two solutions, namely ( x , y ) = (0 , 0) and (0 , − 1) . Proof. Factor the left-hand side: y ( y + 1) = x 3 . Unique factorisation in Z : If gcd( a , b ) = 1 and ab = x 3 , then a = x 3 b = x 3 1 , 2 . Hence y and y + 1 are perfect cubes, { y , y + 1 } ⊂ { . . . , − 27 , − 8 , − 1 , 0 , 1 , 8 , 27 , . . . } . It follows that y = − 1 or 0.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT First example: the equation y 2 + y = x 3 Theorem The equation y 2 + y = x 3 has only two solutions, namely ( x , y ) = (0 , 0) and (0 , − 1) . Proof. Factor the left-hand side: y ( y + 1) = x 3 . Unique factorisation in Z : If gcd( a , b ) = 1 and ab = x 3 , then a = x 3 b = x 3 1 , 2 . Hence y and y + 1 are perfect cubes, { y , y + 1 } ⊂ { . . . , − 27 , − 8 , − 1 , 0 , 1 , 8 , 27 , . . . } . It follows that y = − 1 or 0.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT First example: the equation y 2 + y = x 3 Theorem The equation y 2 + y = x 3 has only two solutions, namely ( x , y ) = (0 , 0) and (0 , − 1) . Proof. Factor the left-hand side: y ( y + 1) = x 3 . Unique factorisation in Z : If gcd( a , b ) = 1 and ab = x 3 , then a = x 3 b = x 3 1 , 2 . Hence y and y + 1 are perfect cubes, { y , y + 1 } ⊂ { . . . , − 27 , − 8 , − 1 , 0 , 1 , 8 , 27 , . . . } . It follows that y = − 1 or 0.

Diophantine equations Cubic equations FLT Pell’s equation Elliptic curves Back to FLT First example: the equation y 2 + y = x 3 Theorem The equation y 2 + y = x 3 has only two solutions, namely ( x , y ) = (0 , 0) and (0 , − 1) . Proof. Factor the left-hand side: y ( y + 1) = x 3 . Unique factorisation in Z : If gcd( a , b ) = 1 and ab = x 3 , then a = x 3 b = x 3 1 , 2 . Hence y and y + 1 are perfect cubes, { y , y + 1 } ⊂ { . . . , − 27 , − 8 , − 1 , 0 , 1 , 8 , 27 , . . . } . It follows that y = − 1 or 0.