## Archive for March, 2010

### Google Public Data

Last week, Google unveiled their new Public Data Explorer. If you’ve ever seen Gapminder, this is very similar.

In fact, Google bought the Trendalyzer software from Gapminder and applied it to data from Eurostat, the US Census Bureau and many other sources: Datasets

For me, the main question with Google is where they’ll next head in their persistent quest for market share and eventually world domination. Chrome, Android, Google Maps, YouTube, all added to their repertoire and threatened the competitors in those fields. When was the last time you used Mapquest?

With Public Data, the main target seems to be Wolfram Alpha with its database of everything from the lethal dosage of caffeine to algorithms for solving differential equations. Granted, Google is mostly sticking to Gapminder’s domain: global population statistics. But with their tendency to obsessively add new functionality (also last week: bike routes on Google Maps!), it’s only a matter of time.

### Bizarre Fact About Factors

For *any* integers B and x greater than 1, where B is not a multiple of any factors of x: There exists a positive integer n such that (B^n)-1 is divisible by x.

In other words: 999….9999 is a multiple of any given number, except for evens and multiples of 5 (base 10). You just need the right number of nines.

In binary, 1111…111 has the same property.

I won’t write out a formal proof, but here’s a simple explanation:

The decimal expansion of 1/x is a repeating decimal.

Therefore, it’s a fraction over B^n-1.

Inverting both sides of the equation, B^n-1 is a multiple of x.

This doesn’t actually work if x is a multiple of 2 or 5 (in decimal), or in general any of the factors of the radix, because the repeating decimal will have a number of zeroes before it. But it works in all other cases.

Example:

99999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999999999999999999999999999999999999999999999

99999999999999999999999999999999999999999999999999 is a multiple of 1,337. That’s 570 nines.

Hey, I never said n had to be small.

EDIT: After poking around Wikipedia and playing with Wolfram|Alpha a bit, it seems that an upper bound on N is x-1. A better upper bound, however, is LCM(a,b,c…) where a,b,c are the prime factors of x, minus one.

1337 = 7*191.

LCM(6,190) is — guess what? — 570.