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Two Primes Make One Square

Age 7 to 11
Challenge Level

Two Primes Make One Square

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:
"I made the square numbers out of cubes and tried taking a prime number of cubes away and seeing if it left a prime number of cubes."

Dina said:
"I wondered whether noticing that 2 is the only even prime number was important."

Shameem said:
"I listed the prime numbers up to 100 and then I listed the squares of the numbers from 4 to 20."

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?

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

Pose the challenge without offering any support at this stage and give everyone a few minutes to think on their own about how they might begin. Suggest that they share ideas with a partner and then give them an opportunity to comment and/or ask questions (although make it clear that you are not asking for ways of approaching the task at this stage).  Encourage other learners to respond rather than you yourself and use this discussion to clarify the task, and perhaps address any misconceptions.

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?

Key questions

What are the squares of the numbers from $4$ to $20$?
Try choosing two prime numbers and adding them together? What will you do now?
Do you think you will be able to make that square number if you had enough time?  Why or why not?

Possible extension

Some children might like to explore all the different ways that each square number can be made. How do they know they have found them all?

Possible support

Many pupils will find that calculators, number squares and multiplication squares will be useful tools.