As part of the publicity around the Fields Medal announcement, the American Mathematical Monthly’s Facebook page pointed out this paper written by Manjul Bhargava in 2000:
The Factorial Function and Generalizations
The reason that this paper caught my attention is that I actually felt like I had a prayer of understanding the ideas that the paper was explaining. I’m sure, of course, that the work of all of the 2014 Fields Medal winners is off the charts brilliant, but when I searched for lectures or papers by them everything but this paper was miles over my head. So many miles, actually, that the factorial function might come in handy if I needed to describe that distance accurately!
One particularly helpful example that Bhargava gives in the paper is on top of the 6th page (page number 788) where he calculates the first six values of the generalized factorial function over the primes. Since he said that it was “an easy matter to compute” these six values, I thought that replicating this calculation would be a fun way to see if I had understood some bits of the paper. After a few times through the paper I finally had understood the ideas well enough to replicate the calculation, so yay! I was also surprised to see that those six numbers (1,1,2,24,48, and 5760) appear in the On-line Encyclopedia of Integer Sequences only once (and without reference to Bhargava’s result):
Is this the full generalized factorial function over the primes?
I don’t know if the full sequences would match each other (or, obviously, why that sequence would arise from the Taylor series of ) but at least my calculation of the next term in Bhargava’s sequence does match the next term in the OEIS example.
So, having (hopefully) understood how to calculate this generalized factorial function over the primes, I wanted to try it for another sequence of integers. The Fibonacci numbers seemed like as good a place as any to start, so I gave that a shot this weekend. Unluckily I was traveling this weekend, but still found a little time early this morning at the Lone Wolf diner in Amherst, MA to get things going while enjoying their Santa Fe omelette. According to my calculations, Bhargava’s factorial function over the Fibonacci numbers would evaluate as follows:
0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 240
6! = 720
7! = 443,520 ()
8! = 443,520 (yes, the same as 7!. That’s a surprise, but I haven’t been able to see the mistake)
9! = = 103,783,680
10! = = 1,037,836,800
[edit note: after publishing earlier today, I noticed that I left of the 13 on both 9! and 10! ]
Fingers crossed that these are the correct calculations but since I slept at a farm last and was woken up early by roosters, it wouldn’t be super surprising if there was an error 🙂 In any case, one interesting thing that I learned playing around with this is that the Fibonacci numbers have an interesting pattern modulo 11. I’m hoping to play around with this and other sequences in the next month. I think there is a really fun project for kids hiding in here somewhere, too.