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Getting Perfect Squares With 4 And 9? (Posted on 2007-09-03) Difficulty: 3 of 5
Can 4*10p + 9 be a perfect square whenever p is an integer ≥ 2?

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (6 votes)

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Solution If p is even | Comment 1 of 3

If p is even then let p = 2q. p>=2 implies q>=1

Then sqrt(4*10^p) = 2*10^q.  The two nearest squares to 4*10^p are 4*10^p + 1 +/- 2*10^q.

With q>=1, those nearest squares are too far from 4*10^p to be equal to 4*10^p+9, therefore if p is an even integer then 4*10^p+9 is never a perfect square.

  Posted by Brian Smith on 2007-09-03 14:24:15
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