Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Survey

Document related concepts

Infinitesimal wikipedia , lookup

Georg Cantor's first set theory article wikipedia , lookup

Large numbers wikipedia , lookup

Real number wikipedia , lookup

Pythagorean theorem wikipedia , lookup

Non-standard analysis wikipedia , lookup

Fundamental theorem of algebra wikipedia , lookup

Proofs of Fermat's little theorem wikipedia , lookup

Transcript

Chapter 2 Hints and Solutions to Exercises p. 18 Exercise 2.1 a. All natural numbers can be written in the form 3k multiples of or 3k 1 , 3k 2 , where k 3 non multiples of 3 some natural number. Step 1: First show that perfect squares can only be of the form 3j or 3j+1. For example (3k 1) 2 9k 2 6k 1 3(3k 2 2k ) 1 3 j 1 where j 3k 2 2k . Do a similar analysis for 3k and 3k+2. Step 2: Assume that a and b in the Pythagorean Theorem are both not multiples of 3. For all cases where this is so, derive a contradiction showing that c cannot be obtained since a 2 b 2 does not give a perfect square. For example, if a 3m 1 and b 3n 1 for some numbers m and n, a 2 b 2 (3m 1) 2 (3n 1) 2 3m 1 3n 1 3(m n ) 2 3s 2 c 2 . Since numbers of the form 3s+2 cannot be a perfect square, there is no solution c. Do this for all other non-multiple of 3 cases for a and b. b. Start by showing that perfect squares can only have the form 5j, 5j+1 or 5j + 4 and proceed Exercise 2.6 Answer is (t 2 2t ,2t 2, t 2 2t 2) . To find this, use the fact that st 2 a 2 c and derive a relationship between s and t s2 t 2 2