I was chit-chatting with my boss, Eric, before work one morning. Somehow we got onto the topic of identifying fraudulent financial data. I told Eric about Benford’s Law, and he speculated about a technique I hadn’t thought of before.

Benford’s Law states that in real-life financial data, the leading digit in any number will be a ‘1’ or a ‘2’ almost 50% of the time (theoretically, 48.7%). Very surprising. If a criminal generated fake data, it’s highly unlikely they would make the leading digit a ‘1’ or a ‘2’ so often.

The real-life financial data I used

I decided to see for myself. I grabbed an old, real-life page of financial data. There were 65 numbers on the page. I manually counted the number of times each digit, ‘0’ through ‘9’, occurred as the first digit, and computed each frequency:

digit   count    freq
----------------------
 0        3      0.05
 1       18      0.28
 2       15      0.23
 3        6      0.09
 4        4      0.06
 5        5      0.08
 6        3      0.05
 7        3      0.05
 8        7      0.11
 9        1      0.02
            
         65      1.00

The leading digit in the real-life financial data was a ‘1’ or a ‘2’ with a frequency of 0.28 + 0.23 = 0.51 — very close to the 0.48 predicted by Benford’s Law. Surprising!

Eric wasn’t familiar with Benford’s Law. Before I explained it, Eric speculated that if a criminal made up fake financial data, they would almost never use two consecutive digits that are the same because that doesn’t appear random. I coined the idea the “Ahlstrom Conjecture”.

It turns out that consecutive digits that are the same are very frequent. Theoretically, in a sequence of 10 random digits, the probability that there is a pair of digits that are the same, such as ’33’ or ’77’, is approximately 0.65 : 1.0 – ((10 * 9^9) / 10^10).

I applied “The Ahlstrom Conjecture” to the real-life financial data. I broke the data up into 32 sequences of ten digits. For real, non-fraudulent data, you’d expect about 0.65 * 32 = 21 of the sequences to have one or more pairs of repeated digits.

sequence     repeat?
---------------------
8667894929      x
8949298949
2917898624
4117654517      x
6545718384
1881615806      x
4135792598
0425980424
4103024712
4714561098
0001040041      x
0739148300      x
0012644146      x
4191092821
1214783251
1734225676      x
7292926499      x
6110928211      x
6499611157      x
8172290880      x
6740213173
1143822133      x
3822133292      x
9236051173      x
4225676749      x
6698151729      x
1650002314      x
7262314726
5441313894      x
9339140000      x
1073916469
6801000000      x

The were repeated digits in 20 of the 32 sequences of digits — almost exactly the theory prediction.

Therefore, I think the “Ahlstrom Conjecture” is correct — in fraudulent financial data, generated by a human, you would find far fewer than 65% of the sequences of ten digits to have consecutive same digits.

To the best of my knowledge, this “Ahlstrom Conjecture” isn’t a generally known technique to identify fraudulent financial data. But it seems to be a practical and useful technique. Of course, analyzing data for randomness is a standard technique, but the specific idea of looking for repeated pairs of digits doesn’t seem to be described anywhere (at least that I could find).

Fascinating!

Generating fake financial data is not too difficult. With the rise of generative AI, creating fake images has become relatively easy. I did a brief review of the topic and discovered that several of the discussions used images of Japanese geishas as examples. I speculate this is because of the heavy makeup and somewhat artificial appearance of real geishas to begin with.

Here are three such images. One is real and two are AI-generated. I would not be able to identify which is which. Can you? Answer below.