The equation x2+y2 = z2

1 Table 5.1 is a table of sums of squares. Using the left hand column of squares as a checklist, determine the location of all the square numbers in the table. The only square which appears that is greater than 2500 is 2704.

2 What are the square numbers which appear in table 5.1 along a line joining 0 to 42 + 32 = 52? Given the first of these could you have predicted the others?

3 What are the square numbers which appear along a line joining 0 to 122 + 52 = 132? What is the similarity between them?

4 Use the fact that 242 + 72 = 252 and 152 + 82= 172 to predict two more square numbers in table 5.1.

5 If x, y and z are positive integers such that x2 + y2 = z2, then (x, y, z) is called a Pythagorean triple. If, moreover, gcd (x, y, z) = 1, then (x, y, z) is called a primitive Pythagorean triple. Identify the seven distinct primitive Pythagorean triples which may be deduced from table 5.1, ignoring the possibility that (x, y, z) ≠ (y,x,z).

6 If (x, y, z) is a Pythagorean triple, is it possible that

• (i) all three of JC, y and z are even,

• (ii) just two of JC, y and z are even,

• (iii) just one of JC, y and z is even,

• (iv) neither JC, nor y nor z is even?

• Either illustrate the possibility, or prove the impossibility.

The equation x2 +y2 = z2

7 If (JC, y, z) is a Pythagorean triple, is it possible that

• (i) all three of JC, y and z are divisible by 3,

• (ii) just two of JC, y and z are divisible by 3,

• (iii) just one of JC, y and z is divisible by 3,

• (iv) neither JC, nor y nor z is divisible by 3?

• Either illustrate the possibility, or prove the impossibility. For

• (iv), make a table of sums of squares in Z3.