Talk:First perfect square in base n with n unique digits: Difference between revisions
Content added Content deleted
(Responded to comment about repeated digits from Nigel Galloway.) |
|||
| Line 170: | Line 170: | ||
:::The residuals base 16 are 1 4 9 16<br>0+1+...+15+16 -> 80 -> 8 therefore no 17 digit perfect square made from digits 0..g in base 17 so searching for one is pointless.<br>101246A89CGFB357ED -> 81 -> 9 therefore may be a perfect square, just as well since you say it is.<br>smallest possible number made repeating 1 is 10123456789abcdefg so you only need to verify that there is no perfect square between 10123456789abcdefg and 101246A89CGFB357ED which contains all the digits between 0 and g to prove that 101246A89CGFB357ED is the smallest.--[[User:Nigel Galloway|Nigel Galloway]] ([[User talk:Nigel Galloway|talk]]) 10:07, 24 May 2019 (UTC) |
:::The residuals base 16 are 1 4 9 16<br>0+1+...+15+16 -> 80 -> 8 therefore no 17 digit perfect square made from digits 0..g in base 17 so searching for one is pointless.<br>101246A89CGFB357ED -> 81 -> 9 therefore may be a perfect square, just as well since you say it is.<br>smallest possible number made repeating 1 is 10123456789abcdefg so you only need to verify that there is no perfect square between 10123456789abcdefg and 101246A89CGFB357ED which contains all the digits between 0 and g to prove that 101246A89CGFB357ED is the smallest.--[[User:Nigel Galloway|Nigel Galloway]] ([[User talk:Nigel Galloway|talk]]) 10:07, 24 May 2019 (UTC) |
||
::::Note that adding a zero to a number doesn't change it's digital root so the repeated digit can not be zero so 10123456789abcdefg must be the smallest candidate--[[User:Nigel Galloway|Nigel Galloway]] ([[User talk:Nigel Galloway|talk]]) 12:45, 25 May 2019 (UTC) |
::::Note that adding a zero to a number doesn't change it's digital root so the repeated digit can not be zero so 10123456789abcdefg must be the smallest candidate--[[User:Nigel Galloway|Nigel Galloway]] ([[User talk:Nigel Galloway|talk]]) 12:45, 25 May 2019 (UTC) |
||
::::: Many thanks for confirming that Nigel. I had in fact realized it was true when submitting my original Go entry but omitted to explain the reasoning and so, as Thundergnat rightly pointed out, it looked like I was using a 'magic number'. You're also right that it's not really acceptable to assume an extra digit for bases 13 and 17 without further explanation so in my latest Go submission I'm justifying this from first principles. --[[User:PureFox|PureFox]] ([[User talk:PureFox|talk]]) 17:08, 25 May 2019 (UTC) |
|||
:::So digital root 9, I think, drawn from the dual-symmetry base 17 cycle for perfect squares of |
:::So digital root 9, I think, drawn from the dual-symmetry base 17 cycle for perfect squares of |
||