All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
Sum of two powers (Posted on 2004-08-12) Difficulty: 4 of 5
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)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution 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
Please log in:
Remember me:
Sign up! | Forgot password

Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (2)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Copyright © 2002 - 2021 by Animus Pactum Consulting. All rights reserved. Privacy Information