Season 3 Episode 1

OOMC Season 3 Episode 1. Sums of squares

Dr Vicky Neale joins the stream for epsiode 1 with a problem for us; which numbers can be written as the sum of two squares?

Further Reading

Sums of two squares

The main question from the livestream was;

Which numbers can be written as the sum of two squares?

Here are some of the conjectures you made in the livestream;

  • A square is a sum of two squares
  • Numbers that are a square plus 1, 4, 9, … are sums of two squares
  • Powers of 2 (or 5) are sums of two squares.
  • The largest number in a Pythagorean triple is a sum of two squares in $\geq 2$ ways.
  • If n is a sum of two squares then so is 2n
  • A number 3, 6, or 7 (mod 8) is not a sum of two squares
  • If n, m are sums of two squares then so is nm.

Can you make any more conjectures of your own? Can you prove (or disprove!) them?

If you like this, you might also like to try adding other numbers of squares (for example, what numbers can you make by adding three squares?). Some versions of this question are easy (which numbers can you make by adding one square) and some are hard – it’s sometimes difficult to guess which is which.

Someone asked if we’re going to publish this research. Unfortunately, other mathematicians who came before us have worked out answers to these problems already. Do not look at these until you’ve had all the fun that you can have exploring – once you’ve seen the full answer, you can’t unsee it! Exploring places that have already been explored before is still fun, like climbing a mountain rather than taking the ski-lift. You learn a lot more about climbing if you don’t take the ski lift, even if you don’t make it all the way to the top.

Answers (don’t look at these)

Complex numbers, primes, and residues

For more complex numbers and primes, see Season 2 episode 3 with Ittihad.

For more on quadratic residues (the remainders that you can get by dividing square numbers by a particular number), see Season 1 episode 0 with James.

Complex numbers, quaternions, and more

When complex numbers came up, someone in chat mentioned more the obscure quaternions and octonions. These are “higher dimensional” versions of complex numbers, which you might be interested in (but you certainly won’t learn about these in A-level or equivalent!)

More Vicky Neale

Vicky has two books that you might be interested in if you liked this episode;

  • Closing the gap is the story of cutting-edge research in prime number theory, with plenty of things for you to investigate yourself along the way.
  • Why study Mathematics? has lots more information about what mathematics is like at university and why you should consider studying it.

Vicky has previously appeared on the OOMC in Season 2 episode 0 with a different problem involving squares.


For a six-week maths summer school with Vicky (and other people!), see PROMYS.
The application is free, and you might find the application problems fun, even if you’re not going to apply.



For a summer school with James (and other people!) and lots of online support between April and December with your university applications (especially Oxford), see UNIQ.

This course is a free programme for UK state-school students, and we prioritise students with good grades from backgrounds that are under-represented at Oxford.


If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at]

Please contact us with feedback and comments about this page. Last updated on 29 Apr 2022 12:07.