Prove n⁴+4 is a composite number for n>1
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Answer:
If n is even, n
4
+4
n
is divisible by 4
∴ It is composite number
If n is odd, suppose n=2p+1, where p is a positive integer
Then n
4
+4
n
=n
4
+4.4
2p
=n
4
+4(2
p
)
4
which is of the form n
4
+4b
4
, where b is a positive integer (=2
p
)
n
4
+4b
4
=(n
4
+4b
2
+4b
4
)−4b
2
=(n
2
−2b
2
)
2
−(2b)
2
=(n
2
+2b+2b
2
)(n
2
−2b+2b
2
)
We find that n
4
+4b
4
is a composite number consequently n
4
+4
n
is composite when n is odd.
Hence n
4
+4
n
is composite for all integer values of n> 1.
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