Question

show that n^4 +1 is a composite for all n > 1​

Answer 1

Answer:

If n is even, n4+4n is divisible by 4

∴ It is composite number

If n is odd, suppose n=2p+1, where p is a positive integer

Then n4+4n=n4+4.42p=n4+4(2p)4

which is of the form n4+4b4, where b is a positive integer (=2p)

n4 + 4b4 = (n4 + 4b2 + 4b4) − 4b2

=(n2 − 2b2)2 − (2b)2

=(n2 + 2b+ 2b2) (n2 − 2b + 2b2)

We find that n4+4b4 is a composite number consequently n4+4n is composite when n is odd.

Hence n4+4n is composite for all integer values of n> 1.

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