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## Re: Oldie but a goodie...

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• ... Let f(n) = 4^n + n^4. 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
Message 1 of 3 , Jul 1, 2009
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Jack Brennen <jfb@...> wrote:

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

Let f(n) = 4^n + n^4.
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)
• ... Ah - memories of http://www.leyland.vispa.com/numth/primes/xyyx.htm Phil
Message 2 of 3 , Jul 4, 2009
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--- On Wed, 7/1/09, Jack Brennen <jfb@...> wrote:
> Quick little prime chestnut for you:
>
> Find all primes of the form 4^n+n^4, where n is a positive integer.

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

Phil
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