Sum of Digits

[davieddy]

Originally Posted by axn1

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

Yes. But representing zero might lead to some confusion

[wblipp]

As always in mathematics, it comes down to the definitions. And mathematicians are always looking at variations in the definitions. In this case, the question is whether the Base b number system is defined as

A. The value represented is sum a_nbⁿ using the digits 0 to b-1.

B. The value represented is sum a_nbⁿ using the minimal set of non-negative digits.

Definition B permits both "Base 1" and "Base Golden Ratio." Definition A does not.

http://en.wikipedia.org/wiki/Golden_ratio_base


Dropping the "non-negative" also permits balanced representations (digits from -b/2 to +b/2) that are important in Prime95 and other FFT multiplications.

[m_f_h]

Originally Posted by Mini-Geek

http://en.wikipedia.org/wiki/Unary_numeral_system
8 (base 10) is 11111111 (base 1).

I agree that this cannot be used in conjunction with "digit sum" since the tickmarks are NOT digits 1 (which follows the digits 0 which is always the first one) but "tally marks" as is explicitely written on that wikipedia page.

But anyway I think it is irrelevant, since the puzzle said "least even ..." so it will always be even in base 1 since of the form 2 x parity(symbol) which is a multiple of 2 and thus even.
Also, from base 3 on the answer is always even (and the same for base 2b and 2b+1 - or not ?)

somehow inefficient PARI code follows:

digitsum(n,b=10,s)=n=[n];while(n=divrem(n[1],b),s+=n[2]);s

leasteven(Bmax,n=1)=until(!n++,for(b=2,Bmax,digitsum(n,b)%2 & next(2));return(n))

t=1; for( b=2,100, print( b,": ",t=leasteven(b,t)))

[henryzz]

what program did u use kevin

[Kevin]

I just hacked together something in Mathematica. Didn't take much time to optimize it, or make it particularly clear. Seeing as the 1-100 case is still running, it might've been worth taking the time to do that.

a=0;For[b=1,a==0,b++,d=0;For[i=2,i<51,i++,c=Mod[Total[IntegerDigits[b,i]],2];If[c==1,d=1;Break[]]];If[d==0,a=1;Print[b]]]

You're testing numbers b in base i, c is the parity of the sum of digits, and a makes the program stop when you get a solution.

[davar55]

Did the value for 100 ever get computed?

[science_man_88]

Originally Posted by Mini-Geek

http://en.wikipedia.org/wiki/Unary_numeral_system
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).

0 = even no ?

the one base that gets in the way is base 2 in base b, for natural b, 2 or 0 is even and no other digits need to be added it's only in base 2 that it messes it up.

[davar55]

Unary should always be represented by digits of value 1 not 0.

This way the value of the representation of any number is obtained by
adding the digits, with the multiplicative value of each place value taken
into account. In unary, every place has value 1 (instead of b^n for other
bases b), so n = 111.....1 = 1*1 + 1*1 + ... + 1*1 represents any positive
integer n. (Zero is not well-represented in unary. No surprise. That's
why we invented decimal and binary.)

Unlike the Peano axioms where zero and successor are used (which is
fine) or Russell and Whitehead's axiomatic construction, defining unary
as a sequence of zeroes just to be consistent with reverse induction
(ternary uses 2,1,0, binary uses 1,0, so unary should use only 0)
would be a "foolish consistency". One (versus zero) is the better choice.

[legendarymudkip]

Originally Posted by davar55

In unary, every place has value 1 (instead of b^n for other
bases b)

Rather, in unary, every place still has value bⁿ, as 1ⁿ = 1 for any n.

For bases 1 to 100, I did a check up to 2³² without finding anything.

[Jayder]

Up to 10^10, I found a few numbers that passed all bases up to but not including base 62. This is the highest I've found.

legendarymudkip, what script are you using and would you mind benchmarking it for me? How long does it take to test 10^10 to 10^10+10^7? I am wondering how terrible my program is.

[legendarymudkip]

For bases 1-62, it took about an hour to get to 10^10.
For bases 1-100, I'd assume about 1.5 hours.

My code is probably quite slow, but I don't know.