Pythagorean Triples and Fermat’s Last Theorem Shobha Bagai Institute of Lifelong Learning University of Delhi
PYTHAGOREAN TRIPLES AND FERMAT’S LAST THEOREM Papyrus Berlin: "You are told the area of a square of 100 square cubits is equal to that of two smaller squares, the side of one square is 1/2 + 1/4 of the other. What are the sides of the two unknown squares?" In modern terms we would express this as: If x 2 + y 2 = 100 and x = 3 4 y , what are x and y ? A modern solution in this form might be 3 4 y " # $ % & ’ 2 + y 2 = 100 ( 9 16 y 2 + y 2 = 100 ( 25 16 y 2 = 100 ( y 2 = 100 ) 16 25 = 64 ( y = 8, x = 6. The Berlin Papyrus 6619, commonly known as the Berlin Papyrus is an ancient Egyptian papyrus document from the Middle Kingdom. This papyrus was found at the ancient burial ground of Saqqara. The problem stated in the Berlin Papyrus suggests some knowledge of what would later be named the Pythagorean theorem. Hieroglyphic transcription of fragment 1 and 3 from the Berlin Papyrus 6619. [ grid/BERLIN_001.htm ] The Hieratic fragments from the Berlin papyrus 6619. [ grid/BERLIN_001.htm ]
INTRODUCTION Pythagoras was a mathematician born in Greece in about 570 BC. He was interested in mathematics, science and philosophy. He is known to most people because of the Pythagoras Theorem that is about a property of all triangles with a right angle. However, h 2 = a 2 + b 2 is true only for right-angled triangles. For acute-angled triangles h 2 < a 2 + b 2 For obtuse-angled triangle h 2 > a 2 + b 2 Note that in any triangle, the longest side h cannot be longer than the sum of the other two sides. So h < a + b . If the two shorter sides of a right-angled triangle are 2 cm and 3 cm, then the length of the hypotenuse is given by h 2 = 2 2 + 3 2 = 13 h = 13 = 3.60555 In the example above, we chose two whole-number sides and found the longest side, which was not a whole number. Is it possible to construct a right triangle whose sides are whole numbers? Many of us are familiar with right triangles of sides 3, 4, 5 or 5, 12, 13. These are called the Pythagorean triples or Pythagorean triads . If the three integers a, b, c have no common factor, then they are called Basic Pythagorean triples . Thus, every Pythagorean triple is a basic or a multiple of a basic triple. For e.g. the Pythagorean triple 9, 12, 15 is a multiple of the basic triple 3, 4, 5. PYTHAGORAS THEOREM If a triangle has one angle which is a right angle then there is a special relationship between the lengths of its three sides h 2 = a 2 + b 2 where h is the longest side called the hypotenuse and a and b are the other two sides. Or The square of the largest side is equal to the sum of the squares of the other two sides in a right-angled triangle. PYTHAGOREAN TRIPLE Positive integers a, b, c that satisfy the relation a 2 + b 2 = c 2 are called the Pythagorean triple.
You've reached the end of your free preview.
Want to read all 11 pages?