Download Primes and Squares Challenge

Primes and Squares Challenge
Can all prime numbers be written as the sum of two
square numbers?
For example: 13 = 4 + 9
ie
13 = 2
2
+ 3
2
Try to find a rule which helps you to predict which
prime numbers can be written as the sum of two square
numbers and which cannot.
Primes and Squares Challenge
Can all prime numbers be written as the sum of two
square numbers?
For example: 13 = 4 + 9
ie
13 = 2
2
+ 3
2
Try to find a rule which helps you to predict which
prime numbers can be written as the sum of two square
numbers and which cannot.
Teachers’ notes

Students will try randomly at first – and that is fine!

After a short while, suggest they use some logic/orderliness when
investigating.

If necessary, give the following hints:

List the prime numbers less than 100 down left hand side;

List all the square numbers less than 100 down right hand side;

See which pairs of square numbers can be added to make a prime
number. Perhaps consider: what happens when you add two odd
numbers, two even, one of each…;

Now make two lists – one of prime numbers which can be written as
the sum of two squares and another of those which do not work. Try
to find out what makes one list different from the other;

Check that your rule works for all prime numbers less than 100. Does
it work for some numbers greater than 100?
Extension

Using the prime numbers which cannot be written as the sum of two
square numbers, can these be written as the sum of three or four
square numbers? Are there any rules you can find to help you predict
these sums?
Solution thoughts to original problem:

Some possibles are 2 = 12 + 22 , 5 = 12+ 22 , 13 = 22+ 32, 17 = 12 + 42…

Some which do not work are 3, 7, 11, 19…

Add 1 to the prime number and then divide by 2. If the answer is even,
the prime number cannot be expressed as the sum of two squares.
Otherwise the prime number can be expressed as the sum of two
squares. This works for all prime numbers.