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Topic: number 40....09 not a perfect square (Read 531 times) |
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gkwal
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number 40....09 not a perfect square
« on: Jul 16th, 2007, 9:28pm » |
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Prove that the number 40..09 (with at least one zero) is not a perfect square.
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towr
wu::riddles Moderator
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Re: number 40....09 not a perfect square
« Reply #1 on: Jul 17th, 2007, 1:29am » |
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For an odd number of 0's it's easy then 40..09 - 9 is a square and (sqrt(40..09-9)+1)2 will be much larger.
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Wikipedia, Google, Mathworld, Integer sequence DB
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Eigenray
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Re: number 40....09 not a perfect square
« Reply #2 on: Jul 18th, 2007, 12:13am » |
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Suppose that 4*10k + 9 = x2. That is,| hidden: | 4*10k = x2-9 = (x+3)(x-3).
Let A=x+3, B=x-3, and let d = gcd(A,B). Since A - B = 6, we must have d | 6. But since AB = 4*10k is not divisible by 3, and A,B are both even (x is odd), we can only have d=2.
Now, we have AB = 2k+25k, with gcd(A,B)=2, and A>B. There are only two ways this can happen:
(1) A = 2k+15k, B=2. But then
6 = A - B = 2(10k-1),
which is impossible.
(2) A = 2*5k, B = 2k+1. Now
6 = A - B = 2(5k - 2k),
or 5k - 2k = 3, which happens only for k=1. But this is just the case 49 = 72.
So there are no other solutions. |
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Barukh
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Re: number 40....09 not a perfect square
« Reply #3 on: Jul 18th, 2007, 2:07am » |
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I love your solution, Eigenray! 
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