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...

This preview shows page 1 - 5 out of 17 pages.

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
Image of page 1
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
Image of page 2
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
Image of page 3
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
Image of page 4
Image of page 5

You've reached the end of your free preview.

Want to read all 17 pages?

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture