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Topic: Repunits non-square (Read 334 times) |
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NickH
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Repunits non-square
« on: Jul 28th, 2004, 2:14pm » |
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Show that no number in the sequence 11, 111, 1111, ... is a perfect square.
(Edited as per towr's objection, below!)
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| « Last Edit: Jul 28th, 2004, 3:08pm by NickH » |
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towr
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Re: Repunits non-square
« Reply #1 on: Jul 28th, 2004, 2:40pm » |
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Shall we exclude 1 from that sequence?
Or make it, there is not more than one number in the sequence that is a perfect square..
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| « Last Edit: Jul 28th, 2004, 2:51pm by towr » |
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Eigenray
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Re: Repunits non-square
« Reply #2 on: Jul 28th, 2004, 4:51pm » |
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Good point towr. It's things like this we really need a mod for.
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Grimbal
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Re: Repunits non-square
« Reply #3 on: Jul 28th, 2004, 7:21pm » |
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::
Think of it (mod 20):
All the numbers 11, 111, 1111, ... are == 11 (mod 20), so we search a square that is == 11 (mod 20).
To end in 1, it must be the square of a number ending in 1 or 9.
1[sup2] and 9[sup2] are not 11 (mod 20).
11[sup2] and 19[sup2] don't need to be tested, since they are the same as 9[sup2] and 1[sup2].
::
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| « Last Edit: Jul 28th, 2004, 7:22pm by Grimbal » |
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