#68Open
The factorial-denominator series
Is the series sum_{n >= 2} 1/(n! - 1) irrational?
Plectis · open-source mathematics
I’m building Plectis, an open-source mathematics project. We use AI and Lean to work on eight open problems, and publish the results, proofs and unfinished work so others can continue it. I direct the work and am responsible for the released claims.
The research and how to join
Choose a problem to see its question, results and remaining work. You can read everything in your browser.
Contributors receive credit for their work. If you solve a problem, the result and credit are yours and your collaborators’. How credit works
How this project works — Problem-Sized Lean Worlds
How to contribute — From Spare Compute to Cumulative Mathematics
In numerical order. Each short note describes one problem; the longer records retain additional working context.
#68Open
Is the series sum_{n >= 2} 1/(n! - 1) irrational?
#243Open
Under a rapid-growth hypothesis on an integer sequence, does rationality of its reciprocal sum force the sequence to satisfy the Sylvester recurrence eventually?
#249Open
Is the binary Lambert series sum phi(n)/2^n irrational?
Short note · Results and remaining work · Long working record
#251Open
Is the dyadic series sum p_n/2^n over consecutive primes irrational? Equivalently, is the corresponding consecutive-prime-gap dyadic series irrational?
#257Open
Is the sum of 1/(2^n-1) over every infinite set of positive exponents irrational?
Short note · Results and remaining work · Long working record
#269Open
For a finite set of at least two primes, is the sum of reciprocals of the running least common multiples of the smooth numbers irrational? This library treats the three-prime case.
#1041Open
For a monic polynomial whose roots lie in the open unit disc, must two roots be joinable by a curve of length less than two inside the open unit lemniscate?
#1049Open
For which rational bases is the corresponding series irrational? The smallest resistant explicit base is three halves.
Lean checks formal statements. It does not establish novelty, significance or peer review.
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The file supplies the public files; reading it does not run the checks. A claim that a check passed needs evidence from actually running it.
I am 22, an economics undergraduate at Bristol. I built this alone over about a year and paid for it myself, and AI was the only tool I used.
The eight problems were picked because they are deep, and I did not expect to solve them. The experiment was to see what happens when you throw as much as you can at problems like these, and to build the infrastructure off the failure modes today's models actually show. I designed the system for coding agents to work in, on the assumption that it gets more useful as the models improve. I have not shown that; it is still an assumption. Most of the mathematics is written in Lean, which checks every step, so you do not have to take my word for those claims. The Erdős problems the work covers are open, and none of it solves one.
It is published so that anyone can continue it. The corpus, the tooling, and the open step each paper names are all public: clone them, run your own models at the problems, or improve the infrastructure itself. A solution found that way is yours, and so is the credit.
Nobody independent has checked the mathematics yet, and nobody outside has seen the whole system. If you can evaluate work like this, the most useful thing is bounded: take one representative public claim, follow it to its source and its evidence, and tell me where the wording goes further than the record does. If you are in a position to look at all of it, I would like to show you. I am also looking for work or funding; a fellowship or something like it would let me do this properly instead of alongside a degree.