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        riddles >> easy >> Repunits non-square
        
(Message started by: NickH on Jul 28^th, 2004, 2:14pm)

Title: Repunits non-square
Post by NickH on Jul 28^th, 2004, 2:14pm

Show that no number in the sequence 11, 111, 1111, ... is a perfect square.

(Edited as per towr's objection, below!)

Title: Re: Repunits non-square
Post by towr on Jul 28^th, 2004, 2:40pm

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..

Title: Re: Repunits non-square
Post by Eigenray on Jul 28^th, 2004, 4:51pm

Good point towr. It's things like this we really need a mod for.

Title: Re: Repunits non-square
Post by Grimbal on Jul 28^th, 2004, 7:21pm

::[hide]
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].
[/hide]::