The write-up is so far my favorite out of all of my solutions, because the model sequentially bring in a lot of clever tricks to prove the end-results, namely that the functions at hand are asymptotically multiples of the iterated logarithm function. 2/n

I fully solved my 2nd Erdős Problem using ChatGPT-5.5-Pro - and then I verified the solution by formalizing it!

Less than 2 days after solving my first Erdős Problem, after running Pro for a few hours I was able to elicit the solution, this time in analytic number theory! 🧵1/n

I don’t know - will the solution to the Riemann hypothesis just get posted by some Satoshi-Nakomoto-like figure on some forum in 2028? Actually, there could be no human at all behind such an anonymous account. This doesn’t seem that impossible anymore. 8/n

I encourage you to check it out - it honestly still seems strange that a model was able to come up with all this end-to-end. 3/n

https://www.overleaf.com/read/cmypbrpwbwrv#3e136c

Because ChatGPT-5.5-Pro was a tangible step better at math than 5.4-Pro, many were playing around with it in late April, and somebody had already found, through similar methods, some upper and lower bounds similar to the ones I discovered. 5/n

Going by the pseudonym of ‘Treasure42’, this person was anonymous! I think this hints a bit at how collaboration and communication in research should work in a post-AGI world, where novel results take little human involvement and also little time, yet a lot of compute: 6/n

Because my solution was quite long, the owner of the Erdős Problems forum (@thomasfbloom) found it useful to understand the solution's strategy by reading the solution of Treasure42, and, based on the formalization in Lean, trusting my more complete solution by extension. 9/n

If different initiatives don’t communicate they’re working on the same problem, they might waste compute in parallel squeezing the same result out of the model! And, you might get anonymous researchers, who prefer to contribute but stay under wraps. 7/n

For this result, a weird thing happened when I went to report my findings on the Erdős Problem forum - somebody had already reported a thorough partial solution the day before! 4/n

@thomasfbloom This, again, is a bit telling of the future: we’ll find proof sketches and partial solutions to be paramount to our understanding if they make the cognitive load tolerable in the face of the sheer volume of innovation AI can generate in the long run.

Onto the next one! 10/n

Find the #696 Erdős Problem forum thread here: https://www.erdosproblems.com/forum/thread/696

The write-up here: https://www.overleaf.com/read/cmypbrpwbwrv#3e136c

The (auto)formalization in Lean 4 here: http://github.com/davidturturean/erdos-696/

@DavidTurturean I have a strong feeling that the greatest value in LLM's for science is the ability to make connections between disciplines

Humans physically can't, due to short lifespan and how long specialized knowledge takes to acquire

Imagine having the knowledge of a Ph D in every subject

@DavidTurturean single handedly solving maths atp.

@DavidTurturean How many scientific breakthroughs or incremental bits of progress are sitting there waiting for someone (or something) to make the right connection/analogy?

@DavidTurturean Congrats!

@DavidTurturean very cool need to take a look

@DavidTurturean what was your strategy with 5.5 pro? api, /goal?, codex etc?