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## Two Primes Make One Square

Try with the squares of the numbers between $4$ and $20$.

Did you find any square numbers which cannot be made by adding two prime numbers together?

### 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 familiar with their properties and also discover some interesting number facts. This investigation will encourage children to be persistent as they work at
mathematics and it gives you an opportunity to highlight different methods of approach.

Possible approach

As a starter, ask the class to find all the different $3$-digit numbers they can make from the digits $1$, $3$ and $5$, then to talk in pairs about which are prime. This should bring up some divisibility rules and remind pupils of the properties of prime numbers, so that in fact they will conclude that none of them are prime. You could also use the

interactive teaching programme number grid (developed as part of the National Numeracy Strategy in England) to discuss prime numbers further by, for example, shading all the numbers that are multiples of $2$, $3$, $5$ etc.

Pose the question and if necessary discuss square numbers. You may like to ask pupils to try a few on mini-whiteboards as a start to check that they understand the task, and then leave them to explore the challenge. It would be worth saying that you are not expecting learners to find solutions straight away - they may need to persevere!

A mini-plenary after $10$ minutes or so could focus on different ways of approaching this task. Some children might make lists of the square numbers first and work that way, others might start with pairs of prime numbers. It would be worth sharing these different strategies and drawing attention to any useful systems that the pupils have developed, for example starting with the smallest
square number and working upwards.

As they work, you could invite children to write up their sums on the board or on a wall for everyone to see. This would give a focus for a plenary as you can talk about 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 $20$?

What happens when you add two prime numbers together?

What happens if you add two different prime numbers together?

Why can't you make that square number?

### 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 them all?

Possible support

Many pupils will find that calculators, number squares and multiplication squares will be useful tools. You could encourage children to make a list of the first six square numbers and the primes up to $36$.