9:09 PM
I'm profusely sorry for taking so long to reply to you fgrieu. It was an extremely busy time for me, especially since Matter Modeling SE had just entered Public Beta and it was the most hectic time possible for the site (which I initially had written the proposal for, and had spent 2 years trying to get it approved by the StackExchange company: one failed attempt in 2018 and one successful attempt from 2019-2020).
Also the way you wrote about my paper has been something bothering me for some time, so I figured it might not be an easy conversation to enter. Although now that I've actually been able to read the questions properly (my brain was too cluttered back in 2020!), I see they actually aren't too bad at all.
> "I see that your approch uses "The nuclear spins of the atoms in diethyl-fluoromalonate". Does that mean there is a link between the precise numerical value of the number factored and that particular molecule?"
Barely. There's the link in that diethyl-fluoromalonate can be used to factor 56153, but not much more than that. In order to get the problem of factoring 56153 into a form that can be factored using diethyl-fluoromalonate, one has to do a lot of things. There exists an optimization problem where we construct a function (inputs are binary strings, outputs are real numbers) such that the binary-string input that gives the smallest output, contains enough information to reconstruct the factors.
That optimization problem changes many times, because for example, we have equations like (p + q = 1 + 2z) where (p,q,z) are all either 0 or 1. I think you would agree that we can say z needs to be 0 for that equation to work. Now that we have z=0, the optimization problem changes from being: minimize(p + q - 1 - 2z)^2
To:
Minimize(p + q - 1)^2.
There's quite a lot of this type of work that gets done, before the problem of factoring 56153 becomes equivalent to finding the ground state of some quantum mechanical system associated with the molecule diethyl-fluoromalonate, so I would say the molecule is related to the number, because the number can be factored by that molecule, but the relation does not go much deeper than that.
Many other numbers can have their associated optimization problems manipulated into such a way that they can be factored by that molecule.
About the other question, I think this is the one you're actually more interested in:
> Is there something wrong in the claim that without a new experimental result, the same technique "factors" n=(2^k+11)*(2^k+45) for any k making n a biprime? I can make a new record out of that! E.g. that works for k=31, "factoring" n=4611686138686472687. Does not break my previous "stunt record" in the same vein, though. But if we could generalize to integers which XOR is 100110₂, the sky is the limit.
Let me allow you to catch up on my answer to the first question and make sure I haven't lost you, before I start rambling on for too long! Thanks for your patience and curiosity @fgrieu.
