This morning I was exploring dedekind numbers and decided to take it a little further.

Wrote a bunch of code and came up with an upperbound estimator for the dedekinds.

It's in python, so forgive me for that.

The bound starts low (x1.95) for D(4) and grows steadily from there, but from what I see it remains an upperbound throughout.

Leading me to an upperbound on D(10) of:
106703049056023475437882601027988757820103040109525947138938025501994616738352763576.33010981

Basics of the code are in the pastebin link below. I also imported the decimal module and set 'd = Decimal', and then did 'getcontext().prec=256' so python wouldn't covert any variable values into exponents due to overflow.

https://pastebin.com/2gjeebRu

The upperbound on D(9) is just a little shy of D(9)*10,000
which isn't bad all things considered.

Your Sunday mornings are as wild as your Saturday mornings

You lost me at "numbers"

@MammaNeedHummus

It's a formula thats a *hard* upperbound on the dedekind set. Before that, the best mathematicians out there could at most offer guestimates based on logarithms and guessing the problem complexity along with probabalistic method There was uncertainty before.

Now theres not.

Only caveat is, while the series is general enough and every upperbound for every dedekind so far has been over each respective *known* number, theres no way to tell if that holds.

It does appear the estimate grows faster than dedekind set though, so a soft proof would demonstrate that.

It's still the best method I know of for putting a hard upperlimit on each number in the set. In fact, its the only such algorithm that I know of.

The D(10) estimate sits 12 orders of magnitude above the best *guestimated*/complexity-based estimate.

A bonus is this also offers a method to prove an integer is *not* a dedekind, with 100% certainty (assuming the properties of the known set hold across the whole set).

This is like certain probabalistic primality tests, like miller rabin, where it can determine if something is composite (not a prime) quickly, but needs more tests to increase certainty that a number is probably prime.

Easy to confirm something is highly unlikely to be dedekind, but hard to confirm when it *is* dedekind.

Basically for all known dedekinds, where 'i' is any given value in the set, do:

((i+1)*i)/2/3

If the result is not an integer, but instead has a non-zero floating point component, then the number is definitely not in the dedekind set.

On the otherhand if it *is* an integer, then it *may* be dedekind.

The probability for a given int n, can be derived from the next highest dedekind m above n, along with the density of composites under m, that are mod 6 == 0