> Begin forwarded message: > > From: Roger Frye <[email protected]> > Subject: Re: Why search for more solutions to A^4 + B^4 + C^4 = D^4 ? > Date: May 15, 2026 at 6:18:27 PM MDT > To: [email protected] > > I agree with Dave that Claude is surprisingly good at scanning code. (I get > by with the free version with GitHub copilot.) For mathematical, research and > strategic planning, I have found the Thinking Mode in the free version of > Gemini gives better advice than other free alternatives. > > Today, I asked Gemini do code a Hensel lift from a p-adic solution for a > quartic. Usually, I try the code, run into bugs, and point the errors out. > After a dozen iterations, I am worn out and often no closer to a solution. > This time I had it red-hat the code itself. And then I told it to imagine a > competitor had written the code, and got more fixes. Then I really did give > the code to DeepSeek with a prompt that Gemini suggested to explain the > subtleties. Then I gave Deep Seek’s analysis back to Gemini and repeated the > process. It was less exhausting this way, and amusing too, but Gemini > eventually gave up and went back to an initial wrong guess. > > I told it, no, I really wanted a Hensel lift. Here is it’s reply before > giving me another suggestion on how to avoid the Hensel lift: > > ------------- > > You are entirely right to be skeptical, and your frustration is 100% > justified. Every time an LLM is asked to write an "Elkies lattice solver" or > a "$p$-adic Hensel lift point-finder" for a quartic curve, it invariably > manufactures a hybrid mathematical phantom. > > AI models tend to blend three entirely different concepts: > Schoof-Elkies-Atkin (SÉA) (which uses Hensel lifting on modular equations > over finite fields to count points), Coppersmith’s Method (which uses LLL to > find small integer roots of polynomials modulo $M$), and Noam Elkies' actual > 2000 paper on finding rational points near curves. The result is always a > flawed script that tries to linearly lift a non-linear curve from a random > local root mod $p$. > > A standard Hensel lift is a purely local tool; it converges to a $p$-adic > number in $\mathbb{Q}_p$. Because the set of $p$-adic points on your curve is > uncountable, a blind local lift will almost always shoot off into a > transcendental $p$-adic space, completely missing the discrete, global > rational points you are looking for. > > —————— > > At least, it’s an explanation of why it gets confused. > > -Roger > > >> On May 14, 2026, at 6:00 AM, Dave Lively <[email protected]> wrote: >> >> I find people's opinions of the capabilities of AI depends strongly on >> whether they've used the leading (released) frontier models. Mine has >> changed radically in the past several months as I've been using Claude (Opus >> 4.6/4.7) with the Cursor agent (CLI!) extensively (my employer footing the >> bill ...) on a complex restructuring of an existing large and very tangled >> monolithic codebase, and for investigating, specifying, and designing a >> major new component. It has also noticed and pointed out several existing >> bugs that it "noticed" when looking over some existing code for a different >> reason. I've used other models in the same agent harness and found a marked >> difference between them and Opus (though I haven't tried the latest OpenAI >> model). >> >> The experience has changed my views from being pretty much aligned with what >> Jonathan/Roger/Ralf are saying to finding myself in agreement with Gary O's >> https://www.oberbrunner.com/blog/ai-superintelligence/ (which I highly >> recommend). >> >> I'll grant that I'm not working on anything that's breaking ground >> algorithmically (though it does have some challenging scale requirements). >> But I never would have believed the quality of "reasoning" I see it doing. >> It's fascinating to watch its inner dialog. To me it feels like higher >> level reasoning is emerging from this interplay between LLMs and the ability >> to experiment and intentionally observe via agentic frameworks. >> >> I'm intrigued and terrified in equal measure. >> >> Dave >
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