[FRIAM] Why search for more solutions to A^4 + B^4 + C^4 = D^4 ?

[Roger Frye]

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 <frye.roger at gmail.com> 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 <marcus at snoutfarm.com> wrote:
>> 
>> Here’s some new work by Opus 4.7 with MCP tooling from me.
>> <session_writeup_v9.pdf>.- .-.. .-.. / ..-. --- --- - . .-. ... / .- .-. . / .-- .-. --- -. --. / ... --- -- . / .- .-. . / ..- ... . ..-. ..- .-..
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> 

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