Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 229

all prime numbers

Find all prime numbers

and all positive integers

such that

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: all prime numbers

Well, N = 1 is one answer. For N > 1, N must be divisible by 5, otherwise

would be divisible by 5 by Fermat’s little theorem.

Moreover, it must be an odd mutliple of 5, otherwise

would be even.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: all prime numbers

Got it! Got it!

We will show that

is never prime for N > 1.

If N > 1, it must be an odd multiple of 5, so let

.

Then

It remains to show that

and

are both greater than 1 for all k ≥ 1. Well, they are, because

and

Hence

is a product of two integers both greater than 1 and so cannot be prime.

Thus the only solution is

.

tony123

Member

Registered: 2007-08-03

Posts: 229

Re: all prime numbers

very nice

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: all prime numbers

I’ve found a quicker way.

Let

be an odd integer greater than 1.

Then