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

Home > Just Math
Sum Prime Power (Posted on 2010-12-14) Difficulty: 3 of 5
Each of A and N is a positive integer, and P is a prime number that satisfy this equation:

2P + 3P = AN

Can N exceed 1?

If so, give an example. If not, prove that no positive integer greater than 1 is possible.

No Solution Yet Submitted by K Sengupta    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Possible Solution Comment 1 of 1

Ignore P's primality for a moment.       
1. if P is even, then LHS = {3,7} mod 10; these values of n need not further detain us.       
2. If P is odd, then LHS = 5, mod 10, so we seek divisibility by 5.       
3. Checking small values, it is at once obvious that LHS is singly divisible by 5, unless P(prime or not) = (10n-5). Even then, the power of 5 can at most increase only by increments of 1       
for values n = 5^(m+1)/2 {3,13,63, etc} i.e. where 10n-5 is itself a power of 5. This rate is far less than the increases in P.       
4. Be that as it may, the only prime of form (10n-5) is 5. 2^5+3^5=275 is not a power for n>1, so we are done.

Edited on December 16, 2010, 2:01 am
  Posted by broll on 2010-12-14 15:08:55

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (4)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information