Beyond Binary Claims: Mathematical Discovery Through Human-AI Collaboration Part 16
Fifteen months ago, I started this thread on Substack with (what I thought was) a simple mathematics problem, and desire to find out what help LLMs could provide in exploration of that problem. In those months, the problem was shown to be quite deep, and the increasing capabilities of the LLMs proved progressively more and more useful in plumbing those depths. I have a post in preparation that goes into the mathematics in detail, but in this post I want to give the overall view of where the research led.
By review, the original question asks about the solutions to the factorial equation:
The only solution is \((a, b, c) = (6, 7, 10)\), or 720 x 5040 = 3,628,800. The reason is that the equation can be divided on both sides by \(b!\) giving:
but for a factorial to equal a product of consecutive integers, that product has to be π-complete (see prior post). Also, \(a \ne b\) because no factorial can be a power of another factorial as the exponent of the greatest prime divisor of any factorial number must be exactly equal to 1, and raising to a power would negate that. Since \(c!\) is the product of \(a!\) and \(b!\), and \(b>a\), that greatest prime divisor of \(c!\) must come from \(b!\). But, \(b!\) is divided out of the right side of the equation, so none of the integers, there, that comprise \(a!\) can have that prime divisor. That means none of the integers on the right side can be prime numbers, because if one (or more) were, it (the highest one) would be the greatest prime factor of \(c!\), which contradicts the greatest prime divisor of \(c!\) being in \(b!\).
The result of the above is to restrict the value of \(a!\) to numbers that are π-complete products of blocks of consecutive composite integers. That turns out to be a short list (that can be generated by this program): 72, 90, 210, 240, 420, 600, 720, 1260, 3360, 6480, 9240, 15750, 50400, 117600, 147840, 166320, 194040, 291060, 510510, 970200, 2942940, 4324320, 5762400, 9147600, 17297280, 19136250, 43243200, 85765680, 96049800, 153153000, 15178363200, 37822664880, 401392571580, and \(6! = 720 \) is the only factorial number on the list. Thus the only solution is where \(a = 6\). The deep part of the proof is showing that there can be no other π-complete products of consecutive composite integers than in the above list, but that is left to my upcoming post. This post is about how this work has led to the more general problem of products of factorials that are factorials: the Surányi–Hickerson conjecture.
The Surányi–Hickerson Conjecture extends the above factorial product, and is an unsolved problem of Diophantine equations. It asserts that all the solutions to:
where:
are: 6!x7! = 10!, 3!x5!x7! = 10!, 2!x5!x14! = 16! and 2!x3!x3!x7! = 9!. As in the simple case above, the last two factorials on the left side cannot be equal because that would raise the power of the greatest prime divisor of \(k!\). However, factorials less than \(ℓ_m!\) can be equal (as in 2!x3!x3!x7! = 9!) because those don’t provide the greatest prime divisor of \(k!\).
In the basic case, we divided both sides by \(b!\); here, if we divide both sides by \(ℓ_m!\):
The product of π-complete numbers is π-complete (the one with the longest list of prime divisors will include any prime divisors in an equal or shorter list), so the left side has to be π-complete, and that requires that the right side be, as above, a π-complete product of consecutive composite integers. Thus every combination of factorials on the left side has to be in the list we found above. A simple program can search the list for these possibilities.
What I have just described is what mathematicians would call a “contingent proof” of the Surányi–Hickerson Conjecture. It says, “If the given list is the complete set of all π-complete products of consecutive composite integers, then the Surányi–Hickerson Conjecture is proven true.” In a coming detailed post I will walk through the proof of the completeness of the list, but that will involve some deep mathematics that may be of interest to only a few readers.
Through the last year, I have gone down many paths to find this answer to my original question. The LLMs I used did not hand me a solution, but the discussions with them allowed me to try out proposed paths to a solution. Explaining it to them clarified my own thinking, and often they would come back and tell me where I was wrong, or point me at literature where someone had been there already. This result came from our human-AI teamwork.