Home > Maths, Starters, Teaching > A nice little thought puzzle

## A nice little thought puzzle

Earlier today I listened to episode 19 of my favourite podcast, “Wrong, But Useful (if you haven’t listened, and like maths, then go listen quick!). Every month the hosts Colin Beveridge (@icecolbeveridge) and Dave Gale (@reflectivemaths) set a maths puzzle for listeners to solve.

This month it was Colin’s turn and he set a nice algebra puzzle:

Show that n^4 + 4 is not prime for any integer greater than or equal to two.

Have a little go if you haven’t already….

My thought process went like this:

For even n then the whole things even, but what about the odds?

Could it be solved by induction?

N=3 gives 85. N=5 gives 629. Nothing jumping out there.

I’ll factorise it, see what happens:

After a brief false start where I wrote the square root of 2 is 1 I came up with the solution, which is quite neat.

n^4+4=(n^2-2n +2)(n^2+2n+2)

n^2 – 2n +2 is greater than 1 for all integers greater than 1.

n^2 + 2n +2 is greater than 1 for all positive integers.

Hence n^4 + 4 is a product of two numbers greater than 1 for all integers n greater than or equal to two.

A lovely little midweek puzzle.