Beyond Binary Claims: Mathematical Discovery Through Human-AI Collaboration Part 6
As stated in the last post in this series I thought I was done with this number theory but, "every time I try to get out, they keep pulling me back in!" I sent my ABC Factorial Proof off for publication at a respected number theory journal back in January, and it got through desk rejection and they are searching for a referee to peer review it. So, I decided to discuss it with the new Claude Sonnet 3.7 in case it found new errors. It didn’t and suggested that I look at Brocard’s equation, which is another factorial problem without closed solution. So I did, and it turns out that there is a way to make a small change to the solution for the ABC Factorial case and get the solution for Brocard’s equation (which alluded Ramanujan). Because the referee has not yet been found, the editor is allowing me to add this part giving solutions to three open problems in one proof (ABC Factorial, Primorial Product of Consecutive Integers, and Brocard’s Equation).
These all came from my discovery of a new class of numbers that I call “primorial complete” or “π-complete” for short. As I explained earlier, these numbers have prime factorizations that are not missing (and are thus “complete”) any prime factors up to their largest prime factor. For example, 60 is π-complete because it is
while 20 is not because it is
which is missing 3.
What I discovered was that expressions such as c(c−1) (from ABC Factorial) and (n+1)(n−1) (from Brocard’s equation) stop having π-complete values after the variables become large enough, and because all factorials have to be π-complete, the solutions for those factorials stop as well. This turns out to be more powerful than I could have possibly imagined when I started looking into it.
Sonnet has gotten much more powerful as well in the last six months. It now thinks that my π-complete numbers prove that any polynomial expression will hit these π-completeness limits, so any equation that sets an unknown factorial to them has bounded solutions (as in my triple proof). This is the conjecture I put to it, and we are running code for a Monte Carlo test of this conjecture and working on a general proof. I have plenty of time before publication of the current proof, so I should be ready to follow it up with the generalization.
I am also working on a problem that I have invented for myself. There are “twin primes” where two prime numbers p1, p2 have the property that p1+2 = p2. Mathematicians suspect those go on forever, but there is no proof of that at this time. I’m interested in the cases of twin primes where p1+1 is π-complete. One of those π-complete numbers between two primes is
which is the last before 100,000,000, but do these keep going? No one knows.
In any case, this series of posts does prove that LLMs can be very helpful with advanced mathematics. If I can get the results I have gotten as an amateur, there must be professional mathematicians cranking out major advances around the world with the help of these tools. Bottom line: “Yes, Virginia, machines can do math.”
Number fact: The number 63,700,992 = 2^18 x 3^5 is a cute little (two primes) π-complete number stuck between 63,700,991 and 63,700,993 which are both prime. This sequence of very special numbers starts with 6 = 2 x 3, which is stuck between primes 5, 7.
Because I found a way to map n(n+1) to m(m+k) and proved a π-complete limit for n(n+1) it means I can reverse the map to show that if some k made m(m+k) unbounded, then that contradicts my proof for n(n+1). So that proves there is a π-complete limit for all m(m+k). The value of that limit is k times the highest n for which n(n+1) is missing only the prime factors of k. Q.E.D.