[helpless06]

Can anyone give me any insight as to why n^4+1 cannot be a prime number.

[asterisk_man]

Not sure what you mean. When n=2 n^4+1=17 which IS prime.
Initial investigations (n=1.. 6) indicate that it IS prime if n is an even positive integer.

[cool_dude]

Ok I totally agree with asterisk_man about your question being wrong. But just for fun I will prove what asterisk_man said about if n is an even positive integer than it will be prime. Except I'll prove the opposite which is if n is an odd positive integer than it will not be prime! The proof uses simple mathematical induction. So lets do it step by step

Basis step:
proof this hold for n = 3 so 3^4 + 1 = 82 which is clearly not prime

Induction step:
Assume that it is true for all n up to k. so
(2k+1)^4 + 1= 2k

now prove for k+1
(2k+1)^4 + 1 + (2(k+1) + 1)^4 + 1 = 2(k+1)
(2k+1)^4 + 1 + (2k+3)^4 + 1 = 2(k+1)

We know the right hand side is even so we don't need to worry about it because its 2 times an integer which is obviously going to be an even number and not prime.

The left side after expanding and simplifying and replacing (2k+1)^4+1 with 2(k+1) since that's our assumption (hypothesis) we stated reduces to
2(k-1) + 36k^4 + 118k^3 + 216k^2 + 216k + 82

The result of this is even as well since 2 times an integer is even and all multiples of 2 times some integer are even so the equation must be even. So we proved that for all k odd positive integers the result of n^4 + 1 will be even! NOT PRIME!

I may have screwed up a few of my computations because expanding an exponent to the 4 is not fun and lots of chances for screw up but I'm pretty sure this is the right idea!

[ebaines]

CD: it would have been much, much simpler to simply note that for n odd, n^4 must be odd, and hence n^4+1 is even. Therefore n^4+1 is not prime for all odd n >1. QED.

I must say I don't follow your reasoning at all. What do you mean by: (2k+1)^4 + 1= 2k?? Is the "k" on the left side something different than the "k" on the right? Because surely as you've written it this is not correct.

[cool_dude]

ebaines yes I guess your reasoning works too. Remember there's more than one way to solve every problem. Also have you done much mathematical induction because in mathematical induction you must make an assumption statement that it is true for all k and prove it for k+1. so we replace the equation (2n+1)^4+1 with k instead of n so its (2k+1)^4+1 Now this is where your confusion is coming from... you don't understand the right hand side why its 2k. Its very simply. What are we trying to prove? We are trying to prove that it will always be an even number! What format will an even number look like? It will look like 2 times some integer. That's why its 2k. Its just really a statement I guess because obviously it you start plugging in 1 or some integer for k then left side doesn't equal right side. Honestly you can get rid of that right side 2k if you want and replace it with = even. So it will look like (2k+1)^4+1 = even. If that makes you feel better.