Download Chapter 2 Hints and Solutions to Exercises p

Chapter 2 Hints and Solutions to Exercises p. 18
Exercise 2.1
a. All natural numbers can be written in the form
3k
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