Talk:First perfect square in base n with n unique digits: Difference between revisions
Talk:First perfect square in base n with n unique digits (view source)
Revision as of 20:48, 23 May 2019
, 4 years ago→Space compression and proof ?
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Any conjectures ? [[User:Hout|Hout]] ([[User talk:Hout|talk]]) 17:40, 23 May 2019 (UTC)
:Excellent. This reminded me of [[Digital root]] and [[Casting out nines]]. The Digital Root of a perfect square expressed in base n is a quadratic residual in base n-1. The quadratic residuals in base 9 are 1, 4, and 7. 0 is treated as 9 so 1+2+3+4+5+6+7+8+9 -> 45 -> 9 so there may be a 10 digit perfect square using all the digits in base 10. Just as well since we've found one. So for base 13 the digital root of a perfect square must be 1, 4, 9, or 12. 1+2+3+4+5+6+7+8+9+a+b+c -> 60 -> 6. So any time spent looking for a 13 digit perfect square using all the 13 digits in base 13 has been wasted. I can also determine which digits to repeat when looking for a 14 digit perfect square. To check 1+0+2+5+4+7+7+3+c+a+8+6+b+9 -> 67 -> 10 -> 1.--[[User:Nigel Galloway|Nigel Galloway]] ([[User talk:Nigel Galloway|talk]]) 20:48, 23 May 2019 (UTC)
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