Andy's lecture last week on credit card numbers, and Brad's recent post about drivers licenses got me thinking about the different ways Number Theory can be used to detect different types of fraud. (Like for example, how to detect a fake ID, or a false credit card).

I found an interesting article called "Fraud Detection and Number Theory" written by Declan Laville that gives an example of this using something called Benford's Law (http://www.the-actuary.org.uk/pdfs/05_05_08.pdf).

Benford's Law goes something like this: If you take a collection of statistics, and categorize them by their first digit, there will be significantly more numbers that start with lower digits, than there will be starting with higher ones- which is pretty strange compared the normal distribution you would probably expect. The exact break down Benford discovered is found in the chart below, where n is the first digit of each number (22,000 studied altogether) and P(n) is the percentage of times it showed up. (Interestingly, the formula for P(n) is very close to Log10(1+1/n)).

n P(n)

1 30.1%

2 17.6%

3 12.5%

4 9.7%

5 7.9%

6 6.7%

7 5.8%

8 5.1%

9 4.6%

So how is Benford's Law being used to detect fraud? Here is an "typical" example they give in the article in regards to analysis of expense payments: "An employee with a limit of, say, £100 for unapproved expenses might try to fool the system by putting in a large number of fraudulent expense claims of, say, £95. A Benford analysis will show the actual frequency of expenses with leading digit ‘9’ will exceed the expected 4%–5% level".

Does anyone else know of other ways in which Number Theory is used to detect Fraud?

Or how it has been used in the past, like in the movie Catch Me If You Can? :)