Sums of two squares

1 Determine the positions in table 1.1 of the integers less than 100 which appear in table 5.1.

2 Which columns of table 1.1 contain numbers from table 5.1 and which do not?

3 Which columns of table 1.1 contain odd prime numbers from table 5.1 and which do not?

4 Make a table of sums of squares modulo 4 and prove that no integer of the form 4k + 3 can be a sum of two squares.

5 Make a list of those numbers less than 200 in tables 5.1 which have a factor 3. Determine, in each case, the highest power of 3 which divides the number.

6 Make a table of sums of squares modulo 3. What can you deduce about x and y if x2 + y2 ≡ 0 (mod 3)? And what does this imply about the number x2 + y2?

7 Make a list of those numbers less than 200 in table 5.1 which have a factor of 7. Determine, in each case, the highest power of 7 which divides the number.

8 Make a table of sums of squares modulo 7. What can you deduce about x and y if x2 + y2 = 0 (mod 7)? What does this imply about the number x2 + y2?

9 Suppose a prime number p, different from 2, divides a number of the form x2 + y2, so that x2 + y2 = 0 (modp); then if y ≢0 (modp), there is an integer a such that ay = \ (mod p), so that ﹛ax)2 +1 = 0 (mod p). Thus ax is an element of order 4 in Mp. What does Lagrange's theorem on subgroups now tell you about p? (Compare with q4.18 and q4.19.)

10 Try formulating a contrapositive to q 9 with a view to generalising q 6 and q 8.

11 Explore the validity of the argument in q 9 when p = 2.

12 If x2 + y2 is divisible by 27, must it be divisible by 81? Generalise your argument.

13 Examine the prime factorisation of numbers in table 5.1 with a view to making a conjecture about which positive integers may be expressed as a sum of two squares.