- --- In primenumbers@yahoogroups.com,

Jack Brennen <jfb@...> wrote:

> Find all primes of the form 4^n+n^4,

Let f(n) = 4^n + n^4.

> where n is a positive integer.

If n is even, then so is f(n).

If n is odd, then

f(n) = (A(n) + B(n))*(A(n) - B(n))

with

A(n) = 2^n + n^2

B(n) = 2^((n+1)/2)*n

Hence only f(1) = 5 is prime.

Not also that 5|f(n) if n is odd and coprime to 5.

David (per proxy Léon François Antoine) - --- On Wed, 7/1/09, Jack Brennen <jfb@...> wrote:
> Quick little prime chestnut for you:

Ah - memories of http://www.leyland.vispa.com/numth/primes/xyyx.htm

>

> Find all primes of the form 4^n+n^4, where n is a positive integer.

Phil