Just posted this on the intro site, but think I should have posted here. Oops

At 3am this morning, my mind was buzzing so in order to stop my brain wandering all over the place, I thought about a board game I like playing called Settlers of Catan.

The game (which is very good) has 19 small hexagons that are positioned to form a larger hexagon. It got me thinking about how many smaller hexagons I would need to create an even larger hexagon. I realised that the number of smaller hexagons would always be a prime number and that:

For any odd natural number n,
n squared - (n squared - 1) / 4
is always prime.

Does anyone know if this is true? And if so, is there a proof?

As said above, if this is totally wrong or just plain obvious, please feel to let me know!

 

No, it is not always prime. For example for n=11, 15 or 17 the corresponding number is not prime.

For n=11 it is 91=7x13.