MATH_135_19_FLT_Intro_Beamer_Student - Objectives History of Fermats Last Theorem Pythagorean Triples MATH 135 Faculty of Mathematics University of

# MATH_135_19_FLT_Intro_Beamer_Student - Objectives History...

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Objectives History of Fermat’s Last Theorem Pythagorean Triples MATH 135 Faculty of Mathematics, University of Waterloo Lecture 19: Introduction to Fermat’s Last Theorem Faculty of Mathematics, University of Waterloo MATH 135 Objectives History of Fermat’s Last Theorem Pythagorean Triples Content Content 1.Provide an historical introduction. 2.Define gcd(x,y,z). 3. State: If x, y and z are integers, not all zero, and gcd ( x , y ) = 1 , then gcd( x , y , z ) = 1 . 4.Define aPythagorean tripleandprimitivePythagorean triple. 5. State and prove: Let d = gcd( x , y , z ) . The three integers x, y and z are a Pythagorean triple if and only if the three integers x 1 = x / d, y 1 = y / d and z 1 = z / d are a Pythagorean triple. 6. State and prove: If x, y and z are a primitive Pythagorean triple, then x, y and z are relatively prime. 7. State and prove: If x, y and z are a primitive Pythagorean triple, then one of the integers x or y is even and the other is odd. 8. State and prove: If ab = c n and gcd( a , b ) = 1 , then there exist integers a 1 and b 1 so that a = a n 1 and b = b n 1 . Faculty of Mathematics, University of Waterloo MATH 135 Objectives History of Fermat’s Last Theorem Pythagorean Triples Diophantus’ Arithmetica Pierre de Fermat (1601 (?) – 1635) was a brilliant French mathematician. It was his habit to make notes in the margins of his books and one such note is famous. Fermat possessed a copy of Bachet’s translation of Diophantus’ Arithmetica . Problem II.8 of the Arithmetica reads Partition a given square into two squares. Diophantus did not require the squares to be integers so we might write Problem II.8 as For what positive rational numbers x , y and z is the equation x 2 + y 2 = x 2 satisfied? Faculty of Mathematics, University of Waterloo MATH 135 Objectives History of Fermat’s Last Theorem Pythagorean Triples The Note Adjacent to Problem II.8, and in the margin of his copy of Arithmetica  #### You've reached the end of your free preview.

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