Challenge Level

*This problem follows on from Clock Arithmetic and More Adventures with Modular Arithmetic**.*

Start by considering square numbers in modulo $5$.

$1^2 = 1 \equiv 1 \text{ mod } 5$

$2^2 = 4 \equiv 4 \text{ mod } 5$

$3^2 = 9 \equiv 4 \text{ mod } 5$

$4^2 = ...$

$5^2 = ...$

Continue finding the value of other square numbers in modulo 5 until you notice a pattern and can predict correctly what will come next.

Can you predict the values of $100^2, 101^2, 102^2...\text{ in mod } 5$?

In modulo $7$ we have:

$1^2 = 1 \equiv 1 \text{ mod } 7$

$2^2 = 4 \equiv 4 \text{ mod } 7$

$3^2 = 9 \equiv 2 \text{ mod } 7$

$4^2 = 16 \equiv 2 \text{ mod } 7$

Continue finding the value of other square numbers in modulo 7 until you can predict correctly what will come next.

Can you predict the values of $100^2, 101^2, 102^2...\text{ in mod } 7?$

Investigate the value of other square numbers in modulo $11, 13, 17$...

You may find this power modulo calculator useful!

**When working in mod $p$, where $p$ is prime, how many values do you need to find before you can predict all the rest? Explain your findings.**

Can you predict any values from the very start?

So far you have looked at square numbers modulo $p$, where $p$ is a prime.

**Investigate square numbers modulo $n$, where $n$ is odd but not prime, or where $n$ is even.**

*If you have enjoyed working on this problem, then you may enjoy Euler's Totient Function.*

*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.*