Never Prime

Number Theory Level 3

For integers $n \geq$ googol, which of these numbers can never be a prime?

n^4 + 4^n

\text{Googol } \times n + 1

n

1 solution

Chung Kevin
Oct 5, 2016

First, we will explain why $n^4 + 4 ^n$ is never a prime.
If $n$ is even, then it is clearly a multiple of 2.
If $n$ is odd, then let $n = 2k + 1$ and we have $n^4 + 4^{2k+1} = (n^2)^2 + 4 n^2 4^{k} + 4 \times 4 ^{2k} - n^2 4^{k+1} = (n^2 + 2 \times 4^{k})^2 - (n 2^{k+1 } ) ^ 2.$
This is of the form $A^2-B^2 = (A-B)(A+B)$ , so we have 2 factors > 1. Note: This is also known as the Sophie Germaine Identity

Next, we will explain why the other terms are not valid answers.

$n$
There are infinitely many primes .
Googol $\times n + 1$
By dirichlet's theorem , since $\gcd ( \text{Googol}, 1 ) = 1$ hence there are infinitely many primes in the sequence.

I'm not following your equations. The middle seems to be intended to be the left hand side with an extra term added and subtracted. But the extra terms aren't identical. The subtracted term (in the middle equation) has a (n+1)exponent, where the added term has a (n) exponent.

On the other hand, I can't find a counter-example to your assertion that the value can never be prime. It's a very interesting proposition.

Steven Perkins - 4 years, 8 months ago

There were some errors in the algebra. I've edited it, and used $n = 2k +1$ to make the presentation cleaner. Is this clear?

Chung Kevin - 4 years, 8 months ago

Thanks, very nice!

Steven Perkins - 4 years, 8 months ago

Never Prime

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