Marcus, Very impressive. I have checked your first, smaller solution. At 59 digits, it’s much larger than the ones I have been keeping track of.
> On Sep 29, 2026, at 8:45 AM, Marcus Daniels <[email protected]> wrote: > > 10810393143175239019071959109997663051929470074089157449031^4 > + 22671119786961601373534221417739977913860239743121836709640^4 > + 30582196668976524718377935182763823727592320140313909866640^4 > = 32765458201929412641416034051435520639258357113742031670969^4 > > 1326591068399536779358938892854490273552236453443555002152134071855400^4 > + 4085889211105606765283229362842594602029239643914476858219936731599639^4 > + 6921301304254813490592676315727954784213572880315413059149125052322960^4 > = 7124645438324610608006592007348975282408521144406631588277705329840361^4 > > From: Friam <[email protected]> On Behalf Of Roger Frye > Sent: Monday, May 18, 2026 7:58 AM > To: The Friday Morning Applied Complexity Coffee Group <[email protected]> > Subject: Re: [FRIAM] Why search for more solutions to A^4 + B^4 + C^4 = D^4 ? > > Marcus, > > Some more comments on your collaboration with Claude. > > A Russian team Robert Gerbicz, Leonid Durman, Yuri Radaev, and Alexey Zubkov > implemented a C++ brute force search up to d < 2 * 10^9. They use 2^10 as the > main reduction rather than 5^4 as you and I did. I have also invented a > different type of brute force search on a/8, b/8 instead of on d, c. > > For my elliptic curve research, I have been using sage which includes a > cythonized PARI/GP package. I have also found pari's ellrank and > hyperellratpoints to be the most efficient tools. > > I have seen patterns in the known solutions that lead me to suspect that > modular forms can be used to bridge from a known elliptic curve fiber to a > new one or from a hyperelliptic quartic to a new one. But I have not yet > found any useful methods. > > Feel free to turn your agents loose on my GitHub repository where all of this > is documented: https://github.com/rfryeSigma/Euler_413. In particular, check > out the two tables of known solutions in the code folder: solutions.py, > solutions_uv.csv . These tables include the 13 solutions < 10^27 and also > solution 19 in your table 2. Solutions 14-18 are easily derived from these > tables, but not documented. > > -Roger > > > On May 18, 2026, at 6:08 AM, Roger Frye <[email protected] > <mailto:[email protected]>> wrote: > > Marcus, > Thank you. Very interesting research. > I haven’t found anything False or anything new in it yet, but I will continue > studying the .pdf. > The research misses the recent work by Tito Piezas III, which has proved most > productive. > This could be remedied quickly by including math.stackexchange in your search > corpus. > -Roger > > > On May 17, 2026, at 6:46 PM, Marcus Daniels <[email protected] > <mailto:[email protected]>> wrote: > > Here’s some new work by Opus 4.7 with MCP tooling from me. > <session_writeup_v9.pdf>.- .-.. .-.. / ..-. --- --- - . .-. ... / .- .-. . / > .-- .-. --- -. --. / ... --- -- . / .- .-. . / ..- ... . ..-. ..- .-.. > FRIAM Applied Complexity Group listserv > Fridays 9a-12p Friday St. Johns Cafe / Thursdays 9a-12p Zoom > https://bit.ly/virtualfriam > to (un)subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > FRIAM-COMIC http://friam-comic.blogspot.com/ > archives: 5/2017 thru present > https://redfish.com/pipermail/friam_redfish.com/ > 1/2003 thru 6/2021 http://friam.383.s1.nabble.com/ > > > > .- .-.. .-.. / ..-. --- --- - . .-. ... / .- .-. . / .-- .-. --- -. --. / ... > --- -- . / .- .-. . / ..- ... . ..-. ..- .-.. > FRIAM Applied Complexity Group listserv > Fridays 9a-12p Friday St. Johns Cafe / Thursdays 9a-12p Zoom > https://bit.ly/virtualfriam > to (un)subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > FRIAM-COMIC http://friam-comic.blogspot.com/ > archives: 5/2017 thru present > https://redfish.com/pipermail/friam_redfish.com/ > 1/2003 thru 6/2021 http://friam.383.s1.nabble.com/
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