A large prime number!

Level 2

Find the sum of all integer values of $n$ such that the number $n^4+4$ is a prime number.

The answer is 0.

3 solutions

Victor Loh
Jan 4, 2014

Let $x$ be a value of $n$ that satisfies the equation.

Then $-x$ will also satisfy the equation as any number $a$ squared $=$ $-a$ squared.

Hence the sum of all integer values of $n$ will cancel out to be $\boxed{0}$ .

Look, Victor Loh i think this would have been a better solution for this problem: From Sophie Germain Identity we know that $n^4+4$ can be factored as followed:

$n^4+4=(n^2+2n+2)(n^2-2n+2)$ . But $n^4+4$ is a prime number and therefore one of the factors must be equal to $1$ .Thus, we have two cases: First case:

$n^2+2n+2=1$ $\rightarrow$ $n=1$

Second case:

$n^2-2n+2=1$ $\rightarrow$ $n=-1$ . Thus, $n=\pm{1}$ and the sum is $\boxed{0}$ .

Lorenc Bushi - 7 years, 5 months ago

You made a small mistake, In the FIRST CASE, it should be $n=(-1)$ and in the SECOND CASE, it should be $n=1$ . Correct me if I'm wrong.

Prasun Biswas - 7 years, 5 months ago

Right, Loh, this problem becomes trivial if "integer values" includes negative integers. Otherwise, it's a harder problem. Methinks the problem should have asked for only positive integer values.

Michael Mendrin - 7 years, 1 month ago

Yes I agree. Thanks

Victor Loh - 7 years, 5 months ago

Ahmad Naufal Hakim
Apr 15, 2014

$n^{4}+4$

$(n^{2}+2)^{2}-4n^{2}$

$(n^{2}+2n+2)(n^{2}-2n+2)$

A prime number has only two factors, $1$ and the number itself. So, we have two cases:

If $n^{2}+2n+2=1\implies n=-1$
If $n^{2}-2n+2=1\implies n=1$

Therefore, the sum of all integer values of $n$ is $\boxed {0}$

Prasun Biswas
Jan 5, 2014

Since $n$ is raised to the power 4, if for any positive value of $n$ , we get $n^{4}+4$ as a prime no., also the negative value of $n$ satisfies that $n^{4}+4$ is a prime no. When all the values of $n$ that satisfies $n^{4}+4$ is a prime number are added, the corresponding positive and negative values of $n$ cancel out each other and the resultant sum is $\boxed{0}$

A large prime number!

The answer is 0.

3 solutions

0 pending reports