we have four different drink receptacles.

First, we have a mug from Russia

which is part of a project on supercritical fluids,

my area of research.

Then you probably know I collect plastic water bottles

and Sam Tang just brought me this one from Hong Kong

new sort of water bottle from Hong Kong

they are usually green

Now this is a blue Watson's water bottle

Brady, who doesn't really like my water bottles, brought

me this one from San Francisco, where he films Numberphile

- his maths channel

and if you turn it round, you can see it says

BOXED WATER IS BETTER

I don't agree with him.

And then finally, there is a vodka glass

which I really quite like because it's got my surname,

Poliakov, on it, though it's not spelled quite the same way

Some of you may have seen our video on Vodka

Mendeleyev vodka and of course, Poliakov vodka

So watch it if you haven't.

So for our fourth day of Christmas

we have four drink receptacles

three chemical badges

two periodic table bed covers

and a piece of tartaric acid from a Swiss wine barrel

Let's see what happens for Day 5.

[Dr. Holly Krieger] ...this is known as the Mersenne Sequence.

Well people have heard of the Mersenne Sequence because this famous question that we don't know

how to answer is, "Are there infinitely many prime numbers in this sequence?"

Since this question is so hard, one easier question we could ask would be,

"What do the prime divisors of each element of this sequence look like?"

[Brady Haran] That seems like a harder question!

[Dr. Krieger, laughing] Well it might be a harder question depending on how specific you want to get.