Sum of Digits
 

What's the smallest even positive integer which,
when represented in each base 1 through 50,
has an even sum of digits?
(What about bases 1 through 100?)

:ermm:How do you represent something in base 1? 0, 00, 000???

[quote=axn1;119839]:ermm:How do you represent something in base 1? 0, 00, 000???[/quote]
[URL]http://en.wikipedia.org/wiki/Unary_numeral_system[/URL]
8 (base 10) is 11111111 (base 1). Often written as tick marks to count things as they happen, with separators every 5.
Back to the question, since it must have an even sum of digits in base 1, it must be an even number (not really sure where to go from there...I just noticed this trivial thing and thought I'd point it out).

[SPOILER]13655684[/SPOILER] for 1-50. I'll have the smallest one for bases 1-100 in a few minutes.

[QUOTE=Mini-Geek;119841][URL]http://en.wikipedia.org/wiki/Unary_numeral_system[/URL]
8 (base 10) is 11111111 (base 1). Often written as tick marks to count things as they happen, with separators every 5.
Back to the question, since it must have an even sum of digits in base 1, it must be an even number (not really sure where to go from there...I just noticed this trivial thing and thought I'd point it out).[/QUOTE]

I refuse to consider that as an actual number system with base 1. Sure, it's a fun kind of system, but it has no properties of any "real" positional number system.

[QUOTE=axn1;119851]Sure, it's a fun kind of system, but it has no properties of any "real" positional number system.[/QUOTE]

Just because it doesn't have the same properties doesn't mean it can't fall under the same definition. If you just think of base b as being all words in b letters (give the b letters an ordering, and then order all words lexigraphically, and make a bijection to the integers), then this is the natural notion of what base 1 would be.

[QUOTE=Kevin;119855]Just because it doesn't have the same properties doesn't mean it can't fall under the same definition. If you just think of base b as being all words in b letters (give the b letters an ordering, and then order all words lexigraphically, and make a bijection to the integers), then this is the natural notion of what base 1 would be.[/QUOTE]

Since a "real" base b system utilises digits from 0..(b-1), a base 1 system should naturally confine it to the digit 0 only. It is just question-begging to use the digit 1 and deduce that somehow you get a valid base 1. I mean, base 2 doesn't use 2, why should base 1 get a free pass?

I agree that it is /some/ type of number system, just not a conventional positional number system that is worthy of the "base n" moniker.

[:rolleyes: Ok, so I'm only half serious about this whole thing -- maybe somebody can move all this philosophical musings to someplace else]

It's all a matter of how you define things. If you use a more general definition, base 1 is a very natural concept. If you use the the definition which relies on special properties you only have when b>2 (your "real" number bases), it's no surprise it's not going to extend to the case where you don't have special properties.

[Stupid second question about bases 1-100...taking way too long]

[QUOTE=Kevin;119876]It's all a matter of how you define things. If you use a more general definition, base 1 is a very natural concept. If you use the the definition which relies on special properties you only have when b>2 (your "real" number bases), it's no surprise it's not going to extend to the case where you don't have special properties.[/QUOTE]I don't agree : a base is a well defined concept. In the "base 1" system you described any sign could replace the 1, it is the representation of a number whithout using a base.

Jacob

I think base 1 is the most natural base, as it's used for counting things, but it is only useful, IMO, with the natural set of numbers, {1, 2, 3, ...}.

[quote=Mini-Geek;119883]I think base 1 is the most natural base, as it's used for counting things, but it is only useful, IMO, with the natural set of numbers, {1, 2, 3, ...}.[/quote]
Methinks you have missed the point that the digit in base 1 is 0.
2 doesn't enter into binary, so why should 1 enter into unary?
And "A" doesn't occur in denary (decimal).

David