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Some Powers Sum Square (Posted on 2008-07-04) Difficulty: 3 of 5
Determine all possible nonnegative integer(s) P such that 22P+1 + 2P + 1 is a perfect square.

See The Solution Submitted by K Sengupta    
Rating: 2.3333 (3 votes)

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Some Thoughts it might be final | Comment 7 of 10 |

I try to find a possible integer root for   22P+1 + 2P + 1

Assume:   22P+1 + 2P + 1=( 2p + 2p-1 - 1)^2

let   w=2P

expanding the right side and comparing with

the left side we get 

 4*w=(w^2)/4

w= 0 or w=16

giving  p=4 as the only non-negative solution, w=0 is ignored,since w=2 is always positive.

  p=0  is a trivial solution.

 It  seems (maybe a formal proof is required) that there is no other way to factorize 22P+1 + 2P + 1 into two equal factors

 

Edited on April 26, 2011, 3:36 pm
  Posted by Ady TZIDON on 2008-07-05 01:22:14

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