Question

Prove that x^4 is prime only for one value of x

Answer 1

How do you show that for all n∈N,n≥2,

n4+4 is not a prime number?

My attempt:

I see that whatever number n4+4 makes when n is an even number would result to an even number. Thus it is not a prime since it can be at least divisible by 2.

When I tried n=x3 where x us any integer n>1, then n4+4 would at least be divisible by 5 which is also not a prime number. The x means the number before 3, it does not mean it's being multiplied. The same rule applies for n=x1 and n=x7 where it results in n4+4 being divisible by 5.

Sorry if I'm not explaining with proper mathematical notation.

For any number n=x5 (eg. n=5,n=15,n=25…etc), the number is hard to determine if it's a prime. Eg. 54+4=629, which is sometimes difficult to determine if it's a prime. But I think that any number n=x5 is going to make n4+4 a composite number from the products of prime numbers. (Eg. 54+4=629=17∗37)

Answer 2

Answer:

x^4+4 = x^4+4-4x^2

(x^2+2)^2-(2x)^2

(x^2-2x+2)(x^2+2x+2)

again smaller factor x^2-2x+2=1

x=1

x^4+4 = 5 € prime