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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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Solution Analytical Solution | Comment 8 of 10 |
 22P+1 + 2P + 1  (2P + 1)
=> 22P+1 + 2P + 1 = (2P + 1+k)
=> 22P+1 + 2P + 1 = 2^2p + k+2k+1+2*(2^p)*(k+1)
Up on simplication
=> 2^p(2^p-2k-1)=k(k+2)
if p>0, k should be even, let k=2c
=> 2^(p-2)(2^p-4c-1)=c(c+1)
(2^p-4c-1) < 4*2^(p-2)
=>
Case1:
Let c=2^(p-2)
=> 2^p-4c-1=-1 => not possible
Case2:
c+1=2^(p-2)
=> 2^p-4c-1 = 2^p-2^p+4-1=3
=> c=3
=> if c=3 => 2^(p-2)=4 => p=4
If p=0, 2^(2p+1)+2^p+1=4 which is a perfect square.

The only solutions are p=0,4



  Posted by Praneeth on 2008-07-12 01:56:22
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