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 Sum of two powers (Posted on 2004-08-12)
If n is an integer, show that n4 + 4n is never a prime for n>1.

 See The Solution Submitted by Federico Kereki Rating: 4.2500 (4 votes)

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 Solution | Comment 6 of 7 |

Let F(n) = n^4+4^n

Suppose that n is even. If so, n^4 is divisible by 4 and consequently F(n) is always divisible by 4 and threfore composite....(i).

Suppose that n is odd. Then,

F(n) = n^4+4^n
= (n^2+2^n)^2 - ((2^(n+1)/2)*n)^2
= (n^2+2^n + 2^(n+1)/2)(n^2+2^n - n*2^((n+1)/2)) .....(ii)

Since n is odd, it follows that 2^(n+1)/2 is always an integer.
It can easily be shown that:

n^2+2^n - n*2^((n+1)/2 >= 5, whenever n>=3

Consequently,n^2+2^n +/- n*2^((n+1)/2 >= 5, whenever n>=3

Consequently, combining (i) and (ii), it follows that F(n) is always
composite whenever n>1.

 Posted by K Sengupta on 2007-05-16 11:13:54
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