More elliptic curves for $x^4+y^4+z^4 = 1$?

Solution 1:

Andrew Bremner found the $k$th $m_k$ of small height for $k=8,9,10,11$. Given,

$$a^4+b^4+c^4 = d^4\tag1$$

$$(p + r)^4 + (p - r)^4 + s^4 = q^4$$

where $R_k = p,q,r,s$,

$$R_8 = 6260583580,\; 12558554489,\; -1552770140,\; 11988496761$$

$$R_9 = -1456578618665,\; 2734283895746,\; -639377557145,\; 2452045365504$$

$$R_{10} = -3142543344652846743,\;\, 5557992180974240706,\; -1971111422846551463, 4048310673060768880$$

$$R_{11} = -2361164981843721467350575,\; 62586521087452988953161234,\; 5241104489910083087 0860865,\; 16178554328069755572637088$$

Define,

$$m_k = \frac{4p^2-q^2-s^2}{3p^2-2pq+r^2}$$

Then,

$$m_8 = \frac{233}{60}$$

$$m_9 = -\frac{56}{165}$$

$$m_{10} = -\frac{5}{44}$$

$$m_{11} = -\frac{125}{92}$$

These solutions would be #21, #23, #29, and #30 in Leonid Durman's list, moving Tomita's from #27 to #31. There are now $31$ primitive solutions to $(1)$ with $d<10^{28}$.