Square Difference

This question asks students to investigate the different numbers they can make by considering the difference of two squares and to generalise their results.  They will also need to prove that all numbers of a certain form can be made, and that it is impossible to make numbers of another form.

Students will probably need to use the following information

• Difference of two squares formula $a^2-b^2=(a+b)(a-b)$
• An even number can be written in the form $2k$, and an odd number in the form $2k+1$
• Facts about parity of numbers (for example an odd number multiplied by an odd number is odd)
• Uniqueness of prime factorisation

There are lots of suggestions for support for students in the What's Possible? Teacher notes.

Possible extension

Here is a list of number theory problems.

Students could also consider how they could write all numbers as a difference of two squares if they relax the "integer" condition.

How many ways are there to write any number as a difference of two squares?