AdamJTP

 

So where are we?

I’m looking at repeated sets of 1s that are divisible by the number of 1’s.

So:

111 is divisible by 3

111,111,111 is divisible by 9

Whereas

11 isn’t divisible by 2

1,111,111 isn’t divisible by 7

..and generalized to different number bases.

e.g. 1,111 in base 3 = 27+9+3+1 = 40 which is divisible by 4.

I called these Saturday numbers. The proper name is a Niven Repunit. Niven because it’s divisible by the sum of its digits and repunit because it’s a repeated unit – repeated 1s.

Defining repunits

Okay – in any number base a repunit is:

Where 

is the number base and 

is the number of digits.

And we want:

Which is a fancy way of saying if you divide it by 

you get 0 remainder (so it exactly divides by

). Which is exactly what we’re after.

Enter the mathematicians (Peter and Paul)

@PeterRowlett and @paulhertz swim in a sea of mathematics (whereas I swim in a sea of software development), they know how to do this stuff – and importantly where to look things up.

Paul did a search and found this:

Which looks promising.

Niven Repunits.  That’s exactly what I’m looking for!

Peter did a search and found that Fibonacci Quarterly is at the British Library:

My trip

As it happens, I’m in London on business next week.  I can quite easily make a detour to the British Library to look this up.

But it is helpful to define success/fail criteria before doing things, so these are my criteria for success/failure next week (wrt library trip – success for the business trip will be defined by the quality of the business requirements!).

 

What I hope I find

I hope to get a better understand of numbers, and why (whether?) my original conjectures are true.

What I don’t hope to find

The title of the article is Niven Repunits and

Now if all the article tells me is that Niven Repunits are values of 

that satisfy

then it’s just restating the problem.

 

Why?

 

 

Well if

Then

Where

Let

Then

or

 

Neater (but still obvious)

So saying a repunit is a Niven number (in base b) when:

is really just restating the original problem.

I hope I don’t find this – because even I can already see that!

 

Undue pessimism?

I’m fairly sure that getting something published in any maths journal takes more than a fairly obvious restatement of the problem.

I’m probably reading too much into the title. I don’t expect they publish the question and the conclusion on the same line!

I’ll post an update next week.