Proving Infinitely Many Natural Numbers: Larson 4.1.6

[ehrenfest]

[SOLVED] Larson 4.1.6

Homework Statement

Prove that there are infinitely many natural numbers a with the following property: The number n^4+a is not prime for any number n.

Homework Equations

The Attempt at a Solution

I cannot even think of one such natural number a. :(
I need to find some way to factor this after we put some restrictions on a. That is we need to express a in a special form that makes this factorable. If a is equal to b^4, it is not necessarily factorable. In fact, I don't know of any power of b that will make it factorable. a cannot be a function of n. I really don't know what to do.

 

[sadhu]

a=4
a=(multiple of 5)-1
hence infinite
i guess
correct me if iam wrong
have you any idea of fermat theorm
n^5-n is divisible by 5
can be prooved ,it is a simpler form of fermat theorm
n(n^4-1) certainly n^4 -1 is divisible by 5
add any multiple of 5 to it
you get

 

[ehrenfest]

Fermat's Little Theorem says that if p is a prime number that does not divide an integer n, then $n^{p-1} \equiv 1 \mod p$.

Therefore that will only apply when n is not divisible by 5. We need a proof for all n in N.

 

[durt]

Maybe you can use Sophie Germain's identity:
$$a^4 + 4 b^4 = (a^2 + 2 b^2 + 2 a b) (a^2 + 2 b^2 - 2 a b)$$.

 

[ehrenfest]

Maybe you can use Sophie Germain's identity:
$$a^4 + 4 b^4 = (a^2 + 2 b^2 + 2 a b) (a^2 + 2 b^2 - 2 a b)$$.

Wow. Thanks. I'm glad I posted this question because I never would have thought of that.