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/
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