# Two Primes Make One Square

## Problem

Flora had a challenge for her friends.

She asked, "Can you make square numbers by adding two prime numbers together?"

Ollie had a think.

"Well, let me see... I know that 4 = 2 + 2. That's a good start!"**Have a go yourself. Try with the squares of the numbers from 4 to 20.**

Once you have had some initial ideas, take a look at how three more of Flora's friends started the problem:

Bailey said:

Dina said:

Shameem said:

Did you go about the task in the same way as any of these children?

What do you like about each method?**Continue working on the problem. You might like to adopt Bailey's or Dina's or Shameem's approach.**

**Did you find any square numbers which cannot be made by adding two prime numbers together? Why or why not?**

You could print off this sheet which includes the problem and the three approaches.

## Getting Started

What are the squares of the numbers from $4$ to $20$?

What happens when you add two prime numbers together?

What happens if you add two different prime numbers together?

## Student Solutions

We had quite a few solutions submitted for this task.

Ashna from St George Vazhathope in India sent in the following:

I thought about the perfect squares between 4 and 20. Then I thought about the prime numbers between 4 and 20.

Norwich Lower School pupils Sanjit, Thomas, Julian, Charlie, Thomas, Isobel, Arnie, Jack, Edward, Freddie, Jack, Mateo, Marissa, Jasmine, Isaac, Amelie and Freya sent in solutions. Benji summarised it well saying:

Yes, it is possible, this is what I got;

11+5=16=4 squared

23+2=25=5 squared

13+23=36=6 squared

2+47=49=7 squared

51+13=64=8 squared

2+79=81=9 squared

89+11=100=10 squared

11 squared is impossible

139+5=144=12 squared

167+2=169=13 squared

193+3=196=14 squared

223+2=225=15 squared

So what I did is wrote down all the square numbers then I found two numbers that were prime and added together to equal the square numbers.

Olive, Edith, Daisy, Taylor, Leah and Millie from Tidcombe”¨ wrote:

We couldn't make 1 with two prime numbers.

We could make all the other square numbers to 100. Although there are usually two ways to make all the even numbers, there can only be one way to make 4 with two prime numbers because the only way to make 4 is 1+3 and 2+2. 1 is not a prime number.

If the square number you are trying to make is even, you will find more than one solution.

If the square number you are trying to make is odd you can only find one solution that uses 2 as one of the prime numbers. In all the odd square numbers, one of the prime numbers for the solution is a 2.

However, the square number 1 breaks the pattern and can't be made with any prime numbers. This is because an odd square number can be made from an odd and even prime but 2 is the only even prime number.

Here are the solutions we have found:

2 + 2 = 4

7 + 2 = 9

11 + 5 = 16 13 + 3 =16

23 + 2 = 25

23 + 13 = 36 7 + 29 = 36

47 + 2 = 49

59 + 5 = 64 53 + 11 = 64

79 + 2 = 81

83 + 17 = 100 53 + 47 = 100

Rishi from Orton Wistow Primary School wrote:

4 squared is 16. 16=11+5

5 squared is 25 25=23+2

6 squared is 36 36=31+5

7 squared is 49 49=2+47

8 squared is 64 64=61+3

9 squared is 81 81=2+79

I did it in a similar way to Dina and that was significant because I realised if the square was an odd number the addition would have to involve 2 to achieve the possibility.

I liked the way Bailey did it because it involves looking at the problem visually and not just mathematically.

I like the way Dina did it because it involves noticing patterns and using them to your advantage.

I like the way Shameem did it because she listed the most important numbers so she could use them later.

I did find a square number that couldn't be made in this way, 121, the reason being that, being an odd square number, the addition has to be completed with 2 and 119 (121-2) is equal to 17 times 7 (also 119).

We had these five responses from the International School Brussels. They came from TJ, Emma, Fede, Luna and Raili. Click on the image to open a larger version on a new page.

Thank you for all the solutions that were submitted.

## Teachers' Resources

### Why do this problem?

This problem brings together two important classes of numbers: primes and squares. By working with them in an investigative way, learners will become more curious about their properties, discover some interesting number facts and develop their number fluency.Highlighting different methods of approach will help children build up their mathematical 'toolkit' and therefore encourage them to be more resilient problem solvers.

Possible approach

Invite pairs to make progress on the task and warn them that you will be bringing them together again before they have had time to get to a full solution. As they work, you could invite children to write up their sums on the board or on the working wall for everyone to see.

After a suitable length of time, focus on different ways of approaching this task, either by drawing on methods that children in your class have used, or by using the examples in the problem (Bailey's, Dina's and Shameem's approaches). If doing the latter, you might like to print off and give out copies of this sheet, which includes the problem and the three approaches. Discuss each one and then give learners time to continue the task, changing their approach to use one of the three methods, should they wish.

In the final plenary you could draw attention to the working wall and the square numbers that haven't been made. Are they impossible to make or is it just that no-one has found a way yet? Why?