Every integer of the form n^4+4, with n>1, is composite?

[Math100]

Homework Statement

Establish the following statement:
Every integer of the form n^4+4, with n>1, is composite.

Relevant Equations

None.

Proof: Suppose a=n^4+4 for some a$\in\mathbb{Z}$ such that n>1.
Then we have a=n^4+4=(n^2-2n+2)(n^2+2n+2).
Note that n^2-2n+2>1 and n^2+2n+2>1 for n>1.
Therefore, every integer of the form n^4+4, with n>1, is composite.

 

[fresh_42]

Homework Statement:: Establish the following statement:
Every integer of the form n^4+4, with n>1, is composite.
Relevant Equations:: None.
Proof: Suppose a=n^4+4 for some a$\in\mathbb{Z}$ such that n>1.
Then we have a=n^4+4=(n^2-2n+2)(n^2+2n+2).
Note that n^2-2n+2>1 and n^2+2n+2>1 for n>1.
Therefore, every integer of the form n^4+4, with n>1, is composite.

If you add double sharps then you get automatically the correct form:

Homework Statement:: Establish the following statement:
Every integer of the form $n^4+4$, with $n>1$, is composite.

Proof: Suppose $a=n^4+4$ for some $a \in\mathbb{Z}$ such that $n>1.$
Then we have $a=n^4+4=(n^2-2n+2)(n^2+2n+2).$
Note that $n^2-2n+2>1$ and $n^2+2n+2>1$ for $n>1.$
Therefore, every integer of the form $n^4+4$, with $n>1$, is composite.

Here is my version for comparison:

\begin{align*}
n^4+4&=n^4+4n^2+4 - 4n^2=(n^2+2)^2 -4n^2\\
&=((n^2+2)-2n)((n^2+2)+2n)
\end{align*}
Since $n^2+2n+2 > n^2-2n+2>1$ for all $n>1$, $n^4+4$ has a non-trivial factorization, i.e. is composite.

 

[Math100]

If you add double sharps then you get automatically the correct form:
Here is my version for comparison:

\begin{align*}
n^4+4&=n^4+4n^2+4 - 4n^2=(n^2+2)^2 -4n^2\\
&=((n^2+2)-2n)((n^2+2)+2n)
\end{align*}
Since $n^2+2n+2 > n^2-2n+2>1$ for all $n>1$, $n^4+4$ has a non-trivial factorization, i.e. is composite.

Sorry, but I don't understand 'double sharps'. What does this mean?

 

[fresh_42]

Sorry, but I don't understand 'double sharps'. What does this mean?

Type $a= n^4+4$ instead of a=n^4+4.

 

[Math100]

Type $n^4$ instead of n^4.

Thank you, now I got it!